Style is something that everything has; each anime, each movie has a certain style. Some can be grouped together into categories. The style of an anime is what gives it its distinctive flavor. When you walk away from a James Bond film and think "I could do that!", when you walk away from Sleepless in Seattle feeling you could fall in love, and when you walk away from CJAS with an exhilarating feeling that all is right with the world - that's style.
Do not be deceived that when you look at anime, the genre doesn't dictate the style. Let's have some examples: If you knew CJAS before this year, you'll recall Maison Ikkoku, one of our best-loved series. The genre was, I don't know, say everyday-soap-opera. The style came from the combination of emotions you saw mostly through Godai, and the odd antics and strange occurrences that cause plot twists or add humor. Patlabor is another good example of a genre/style contrast. Labors are mecha, no doubt about it, and the characters are a police force, straight-forward crime-fighting stuff. The style leads away from heavy labor combat many times and has a more somber quality. The Patlabor movies show an obvious style contrast. The first is faster-paced and gives a greater feeling of danger and excitement. P2 is much darker, calculating, and a bit religious.
For me, the greatest deciding factor in whether or not I enjoy watching anime is the style aspects. Bastard! is a good example of this. The plot is bad guys are trying to save the world, good guys release another bad guy to try and save them; eh pretty standard. The characters: Dark Schneider is a cocky, supergodly wizard trying to take over the world, Gara is the stereotypical super ninja master, Arshes Nei is Darshu's old lover-sorceress who is now forced to fight him. It's not all that special by itself. This show's style is evidenced by the attitudes of the characters and the way they execute their roles. Everyone thinks they're the most powerful, they are all indeed very powerful, and everything is named after a heavy metal band. When you say "That's COOL!", you're talking style.
Evangelion is an example where everything comes together. The characters are complicated, the plot has a few turns, and the style is striking. By sticking two very quiet people in the entry plugs of Earth-defending Eva units, the show takes on a more introspective feel. Instead of getting us psyched up to destroy everything in sight (as the all-Asuka channel might do), we are drawn more toward Rei and Shinji's personalities.
Music plays a very important role in setting the stage for the anime, and since every piece of music has a style, it contributes to the style of the anime. The opening theme of Evangelion has a medium pace most of the way, but it speeds up at the end. The lyrics (which I am sure many of you don't know), show off a great deal of style. They're disjointed and hard to translate (so say those who have tried), but they talk about the boy (Shinji) having to become a legend and that he opens his heart to the singer and a lot of destiny-related stuff. The second verse (which isn't in the episode opening), is more romantic; the singer wishes to preserve the boy as he is now, and she says that she will be his guide to his destiny. The faster-paced part at the end goes a bit like: "Shot out from the hot slime (in the Eva plug), fly from the windowsill to shine in the universe." Well that's very rough but something like that.
Point is, if you want to be an Eva pilot, you've been caught up in the "style-effect"!