If you do not want to see any spoilers I recommend that you leave now, as each episode will be summarized. Keep in mind that I have only seen episodes 1-20. The series thus far goes up to 26 (even then the story isn't resolved). Will eventually conclude with the release of Evangelion movies.
In episode 1 we are introduced to Shinji Ikari, a young man at the age of 14. And as he enters the city that will soon be his home, he is welcomed by a titanic monster which nearly steps on him. Luckily he is saved by a woman named Misato. She drives him out of the situation and away from the attack. The monster which is known as an "Angel" then has a big bomb dropped on its head which manages to put a really big hole in the ground, and stops if for a short time. Meanwhile, Shinji is taken to NERV headquarters where he meets his estranged father. He is then told that he has been brought there for one purpose: to pilot the weapon known as Evangelion. Shinji refuses and they bring out the next formidable pilot who is wheeled out on a cart, and full of bandages. She can hardly stand. The angel then attacks nearby and the supports of the headquarters gives way, and would have crushed Shinji if it were not for the Evangelion acting on its own and saving him. Shinji agrees to pilot the Evangelion. So he is stuffed inside the cockpit and sent up into battle. The Evangelion unit 01, which is the mightiest weapon man kind has ever produced, takes one step forward then gracefully falls flat on its face. The angel then decides to "help" the Evangelion up by grabbing its head and punching it. The Evangelion is thrown back, with it's head case cracked.... it's power shutting down...
In Episode 2 We see that Shinji wakes up. He's got a hangover without the partying the night before. Misato decides that he can live with her. So Shinji finds out what a slob Misato is. He decides to take a shower and runs out screaming naked after seeing that there is a penguin in the house. Misato by the way is rather unimpressed by seeing Shinji naked, although Shinji is extremely embarrassed. That night as Shinji lays in his bed, he recalls the battle with the Angel... Something happened during the battle, in which the Evangelion reactivated and began to act on its own. The Evangelion roars in rage and attacks the Angel. The Angel generates an AT field to parry the attacks of the Eva. The Eva then neutralizes the At field, and pounds on the Angel for generating that annoying thing in the first place. The Eva then engages the Angel and attacks it with an onslaught of blows, or simply put "the Eva done whooped that Angel's ass". The Angel self destructs and the Eva survives no worse for wear.
In Episode 3 Shinji enters his training as a pilot, and enters school. It becomes known that Shinji is the pilot and he ends up with a bloody lip after school from a classmate. Then comes the inevitable attack of another angel, and we actually hear Rei say something! Shinji's classmates then decide to go site seeing and check out the Angel. Eva 1 goes into battle and has the umbilical cord severed limiting the power to one minute. The Eva then ends up protecting the students, who eventually end up in the cockpit. Shinji disobeys the order to retreat and defeats the angel.
In Episode 4 Shinji starts feeling sorry for himself and decides to use public transportation for all it's worth. He is then drug back to NERV by agents in black (sinister look mandatary). Shinji then decides to leave the city. But his mind isn't quite made up yet. After all his thinking he decides not to go... and we are forced to look at a frozen frame for about forever.
In Episode 5 We see the test of Unit 0 that sent Rei into the hospital for a while. Then we skip to the present, where Misato tortures.... I mean, cooks for Shinji and Risuko. At dinner Shinji is asked to deliver Rei's new I.D. card. So Shinji complies. The next morning he goes over to Rei's apartment before school. Shinji carefully enters Rei's apartment... and 30 seconds later we see Shinji laying on top of a nude Rei on the floor with his hand on her chest. Shinji being a Romeo? No, Shinji is just being a klutz (although fairly lucky at being where he lands). Of course you'd never know that anything had happened by the expression on Rei's face. Later in the day Shinji manages to get slapped by Rei (he's quite a busy boy!) Then comes the inevitable fight against the Angel. The mighty Evangelion Unit 1 against the giant blue salt crystal! A few seconds later, unit 1 is toasted and the giant salt crystal goes on about it's business ~ which is digging for pesky NERV agents.
In Episode 6 Shinji gets stuck in the hospital after a disastrous attack by a octahedron (see? Geometry really can be hazardous to your health!). So Misato comes up with a plan.... Shinji then decides to feel sorry for himself..[go figure], after he accidently shows Rei a wee bit more than he intended to. But eventually he decides to cooperate. And so the plan is executed... all the power of Japan is used to fire a proton cannon at the attacking Angel. [translation = they shoot the Angel with a big squirrel gun]. So Shinji shoots and misses... the Angel shoots again before Shinji can fire back.. And Rei saves the day in unit 0; shielding him with a shield (sorta borrowed from the space shuttle). Shingi finally takes out the Angel, and Eva 0 goes down. Shinji then opens the hatch for unit 0 with his bear hands. And so Shinji finally gets to see Rei smile...(and neither of them were even naked!).
In Episode 7 we see what Misato considers a traditional breakfast (hollering and drinking beer). Misato then demonstrates her amazing parking ability, and surprisingly no one is hurt. Later Misato and Risuko go to a conference, and Misato decides to beat the tar out of a locker. And during the demonstration of a corporate made mecha; the mecha decides to go on a walk to a nearby city were it will go into nuclear meltdown (sorta like sailors on shore leave, except with more radiation). So Eva 1 is called in to stop it. Misato goes in side it and manages to disable it before it blows. After all... party girls are not so easily thwarted!
In Episode 8 Shinji and his friends go on a tour of an air- craft carrier. And thus we meet Asuka... and a couple seconds later we see the German / Japanese exchange rate for certain viewing privileges. Dinner with Kaji proves to be rather interesting... and Asuka later decides to show Shinji her........... Eva! Then an Angel attacks. Asuka gets ready for battle. Shinji goes drag and joins her in her Eva. Eventually after fighting over who is in control of the Eva, Shinji gets friendly and sits in Asuka's lap; the two work together and manage to shove two battleships down the Angels throat. After the Angel's destruction, the Eva makes a landing, and in typical fashion, goes into shutdown mode in the ever elegant dead ostrich pose. Kaji takes the thing that the Angel was after to Gendo.... some sort of human omelet.
In episode 9 we see Asuka has managed to become rather popular. Asuka then decides to try talking to Rei, which doesn't work out so well. Thus comes the attacking Angel. Evas 1 & 2 are dispatched to take care of it. Asuka makes the first strike and manages to cut the Angel in half.. The Angel then divides into two Angels. A short time later we see that the Evas on the big screen. Unfortunately the camera angles weren't exactly flattering, but then again it's hard to get a flattering camera angle of an Eva when it's head first in the mud. The U.N. then uses their favorite weapon: the N2 mine, and make yet another big hole in the ground (and stalls the Angel). So Misato starts a little plan. Asuka moves in with Shinji and Misato! Shinji and Asuka are then forced to do (almost) everything together. And they drill day after day trying to coordinate an attack pattern. Until the last night when they finally get to sleep apart... sorta, until Asuka mistakenly ends up on Shinji's sleeping mat. Shinji then decides he's "tha man" and tries to kiss Asuka in her sleep.. But he chickens out. Thus comes the day of the attack. The Evas win over the Angels, but the Evas end up stuck in a tangled mess after shut down. We then get to see that the Evas even have cell phones, which Asuka promptly uses to chew out Shinji for bad parking.
In episode 10 we see Asuka buys a new swim suit. Later Shinji is trying to do his homework at the pool side... until Asuka and her new swim suit show up. Asuka tries to help Shinji with physics with an analogy..(gee, I'd probably learn physics faster too if there were more analogies using breasts!). Elsewhere NERV finds a Angel incubating in a volcano. So Asuka is sent diving in a suit that makes her look like a sumo wrestler. Asuka gets attacked by the Angel which hatches early. She manages to win, but her cable is cut and she would have sunk to the bottom of the volcano if Shinji hadn't saved her. So the gang goes to a nearby hot spring where Shinji listens in to Misato and Asuka playing around. Pen Pen then gets to see Shinji's analogy for thermal explanation...
In episode 11 we see what happens when NERV doesn't pay the electric bill, and the electricity gets cut off. So the three young Eva pilots try to get into NERV headquarters. It will take a miracle... especially since Asuka takes charge. And thus the Angel attacks with NERV practically helpless, and with the three pilots lost due to Asuka's wonderful leadership abilities. But eventually Rei manages to lead them on the right path. The Evas then launch manually, and are forced to do battle with an Angel which seems bent on destroying NERV headquarters by drooling on it. Eventually Asuka does come up with a plan that works and the Angel is defeated.
In episode 12 Misato gets promoted, so everyone parties! Surprisingly Misato is the best behaved at her party. All the higher ups leave town and leave Misato in charge, so of course an Angel decides to attack. This time the Angel is a big 'ol blob who flies around in space destroying satellites for fun (not the MTV satellite!). The Angel is intent on falling on top of NERV and squashing them (not the most complex battle plan ever conceived but effective enough). Misato decides to send the Evas out to repel the attack. So on the first down, the Angel makes a charge... the defensive line manages to fend off the Angel and destroy it. Shinji gets praised by his father and Misato takes the pilots out to lunch, which is nearly all of her money (oh no! No more beer money!).
In episode 13 We see a typical check on things at NERV, until the angel shows up. This time it's more or less a computer driven angel. The angel (emulating Windoze 95 ) starts taking over the system (and everyone knows that nothing will work then!) So the plan is to hack the angel. So Ritsuko and the angel go head to head via computer (like playing pong except faster), with Ritsuko winning.
In episode 14 we see a recap (in case you forgot the other episodes). Along with a psychedelic sequence of Rei asking questions.
Ritsuko: "Shinji, how does it feel to pilot Unit 0?"Shinji: "kinda weird... it smells like Rei"Asuka - "It smells like WHAT?
Rei also shish-kabob's the first angel.
In episode 15 we see Kaji has another job (and NOT at McDonalds!). He is a secret agent for the Japanese Government. Shinji makes Rei blush. Misato gets drunk and her lunch checks out. Asuka gets bored... and finds a new way to kill time, and Shinji gets his first kiss by the same method. Later Rei floats around in a tube. And Misato and Kaji take a tour of NERV to see the shish-kabobed angel.
In episode 16, Asuka does the "Yay! Shinji cheer" and beats the crap out of a locker. And poof, the angel appears. This time it's a big zebra ball with a big honking shadow. Shinji jumps the gun and gets sucked into the shadow (which actually is the angel). Shinji, after being alone for hours, goes on a psychedelic trip (no shrooms included). After having a brief chat with himself he sees his mother.
Meanwhile the Eva's power runs out and it acts on it's own. The Eva tears out of the angel. It kinda looks like birth. Is it a boy, is it a girl?? Aww.. It's an Eva! ( I dare any doctor to spank that one on the bottom!)
In episode 17, the U.S. branch of NERV was completely wiped out while tinkering with an experiment on Unit 3 (no Christmas bonus for those guys!) And we see the new dummy plug (which allows an Eva to operate without a pilot) is completed. The class rep also has a crush on the fourth child.. Who is Toji of all people!
In episode 18 the fourth Eva arrives. Unfortunately it is a rather unhappy Eva, and starts blowing stuff up. Rei and Asuka are disabled by the Eva which is reclassified as the 13th angel. Shinji refuses to destroy the angel for fear of hurting the pilot, so the dummy plug is activated. The Eva then starts tearing the other Eva apart. Shinji begs for it to stop, but it doesn't (like PBS pledge drive week). The Eva destroys the other Eva's plug, soon after Shinji learns Toji was the pilot.
In episode 19 Shinji is rather pissed when he gets back to HQ. He refuses to cooperate and makes an idle threat or two until he is knocked unconscious by a flood of LCL in his plug. So then we get another one of those weird dream on the subway sequences. This time Touji is nearby, and probably says it best - "this is weird".
So Shinji is thrown in jail and handcuffed (yes discipline is rather strict in the Ikari family). Shiny decides to never pilot an Eva again. And just when he was about to leave, an angel attacks. Asuka and Unit 2 try to stop it with everything they've got, but are unsuccessful. Thus Unit 0 (Unit 1 wouldn't accept Rei) enters the picture. Rei takes things into her own hands by playing quarterback with an N2 mine, but fails also. So Shinji is waking around (and sees the Eva's defeated), when he meets Kaji. After a brief talk about melons (garden type and Misato's), he goes back to NERV. Unit 1 is almost defeated with Shinji back as the pilot, when the Eva goes ballistic. Thus we se an episode of "Evangelion Wild Kingdom" as the Eva eats the Angel.
In episode 20 we see that although Evangelion Unit 1 is the most advanced weapon ever built, it still doesn't have any table manners. After Christmas dinner with the angel, the Eva takes a nap and gets bandaged up. Unfortunately Shinji has disappeared, and merged with the Eva.
Rei survived, and Asuka has a breakdown at her failure to stop the angel when Shinji did. Shinji's consciousness goes into a psychidelic dream type sequence (subway not included). Meanwhile NERV plans to rescue Shinji by reconstructing him. Then a projection of the Eva's desire to combine with Shinji appear in the forms of nude Misato, Asuka, and Rei asking "Do you want to become one with me?" [ O_O!! ](drool). The Eva tries to get Shinji to release his soul. Then the rescue operation interrupts. The entry plug is ejected by the Eva and the LCL and the fluid that was Shinji is lost.
Misato clasps Shinji's suit and weeps, crying out to give Shinji back. Shinji (inside the Eva) sort of hears this and the Eva more or less, spits him out. With Shinji saved, Misato and Kaji (quite literally) do some undercover work.