A Real and More Detailed Explanation of Anime:
Hair: The multi-colored hair comes from the earliest attempts to turn black-and-white Japanese manga (comics) into anime (animation). In order to differentiate between characters, manga artists decided to go outside the usual aspects of Japanese hair, which basically amount to black, black, black, and black in a slightly different style. Because having five girls with black hair moving around in a little multi-panel black-and-white comic would be confusing, due to the characters looking exactly like one another at different angles, manga artists started shading their characters' hair in various different shades of black and white, many of which did not directly translate into normal human hair colors like black, brown, blonde, or red. Some of the colors looked like they might actually be orange, blue, grey, or purple when translated to color. So during the process of turning a manga into an anime series, that's just what they did. If a girl looked like she would have orange hair, they gave her orange hair. And if the manga artist decided to make the girl's (or boy's) hair defy gravity just to make them stand out a little more or generally do something cool that wouldn't be possible in real life, they certainly weren't going to change the look of a popular character when turning it into an anime, because the anime was meant to be true to the original work.
By now, however, the funky hair colors and styles have just become a tradition, and both manga and anime creators simply do it because they can, even in color manga.
Eyes: Getzburg got this one perfect. Early manga and anime creators thought the big eyes in early American animation were pretty nifty, so they gave their characters similarly large eyes. And by now, much like the case with the hair, it's just become a tradition in Japanese anime and manga.
Uniforms: This one is probably the simplest of them all. Japanese school children, at least the vast majority of them, wear school uniforms. Thus, characters portraying Japanese school children wear school uniforms, too.
Naughty Bits: If you've seen anime such as Turn-A Gundam, Neon Genesis Evangelion, or any other TV series that show male or female nudity on screen, you've probably wondered "Where's his/her nipples/pubic hair/penis?". In most (all?) anime TV series, characters that are shown nude are usually missing some of their naughty bits. Girls would have no pubic hair, as well as breasts without nipples, and guys would have a dark spot over their crotch or quite simply a total lack of a penis. Honestly, the only reason for this seems to be that if these parts aren't showing, the characters aren't technically naked, at least for the sake of censorship issues. In many countries that don't allow full nudity on television, the only real restrictions are that nipples, pubic hair, and penises cannot be shown. So instead of covering the characters up right to that point, the artists instead just remove those parts, while leaving the character completely naked nonetheless. As Japanese censorship has become more and more relaxed, though, I think this can also just be chocked up to tradition. The laws against showing pubic hair have been relaxed and possibly even abolished in Japan, so the disappearing sexual organs in recent TV series like Turn-A Gundam are probably just for the Hell of it.
Sentai: Sentai are groups of roughly five heroes that have similar powers and costumes, and are far more powerful when their powers are combined than they are alone. The sentai show that Americans would most readily recognize is the Power Rangers, which are a collection of several live action sentai shows edited into one continuous plot. Power Rangers is not anime because it's a live action show, but it is part of a trend that stretches directly into anime, giving birth to shows like Ronin Warriors (also called Samurai Troopers), Gundam Wing, Sailor Moon, and the Ginyu Force characters in Dragon Ball Z (which were a parody of sentai shows). While to some this may seem like a lot of different shows with the same basic plot, "ripping off" each other almost endlessly, the shows I listed are actually very, very different, which makes sentai pretty similar to American movies and TV series about vampires. Vampires, like sentai, are a very old plot device that have been done a thousand times over... but each new show brings in their own unique little twist on the idea, giving birth to very different shows such as Buffy the Vampire Slayer, Forever Knight, and Ultra-Violet.
And finally, the last thing that English speaking anime fans probably need an explanation about:
Christian Symbols: For many people, the Christian symbolism in shows like Neon Genesis Evangelion, Big O, and Digimon Tamers/Digimon Season 3 (to a lesser extent) has been one of the most persistently confusing aspects of anime, mostly because it just doesn't seem to make any sense. Many fans are left completely dumb-founded by the combination of more Christian symbolism than they have ever seen in Western TV shows, even on EWTN, and the complete lack of a preachy Christian storyline or overt Christian values in the storyline of the show itself. The answer is actually surprisingly simple. It's just a bunch of Japanese people having fun with a religion that isn't theirs. That's really all it is. Much like the way the movie The Mummy treads all over Egyptian myth or TV shows like Hercules: The Legendary Journeys and Xena: Warrior Princess tread all over Greek myth, all three without any display of the true values of the culture, anime treads all over Christianity because it's a safe set of religious myths and legends that they can play with, due to the lack of any large Christian movement in Japan. The key difference between the use of religion in Western and Japanese media, however, is that the Japanese feel safe using active religions, while Western media makers are too afraid of lawsuits to use an active religion in their own very diverse cultures. If Japanese TV companies use Christianity in their shows, there aren't a lot of Christians in Japan to be offended. But if Western TV companies used Christianity, Islam, Judaism, Hindu, or really any other active religion on the face of the planet, there would be a lot of people from that religion in their country to be offended, especially in the US. This is why TV and movie companies in the US tend to stick to dead religions, such as Greek, Egyptian, Roman, Norse, and assorted European mythologies.