>>217
じゃない is the plain form of ではありません (is not). In a question sentence, it is pretty similar to the English equivalent "is not," or rather, "isn't."
Thus 俺じゃない。 It is not me.
猫じゃないか。 Isn't it a cat?
This question can be shortened to 猫じゃん
You also may hear じゃない become じゃねえ if a guy trying to sound cool is talking.
Now to explain んじゃないですか. One way of making negative plain forms (じゃない) polite in casual conversation is to use the negative plain as usual, but tack on です. For example 食べないです or the more casual 食べないっす. Thus じゃないです is grammatically equivalent to ではありません.
The ん before it all is a shortened form of の particle, which in this sentence is difficult to explain, but it kind of "emotes" the sentence, linking the speaker to it, and also hinting that they already know the answer to their question, kind of like when buddies as their friends, "Hey, isn't she hot!" It's a statement and a question put together. It also serves to nominalize a preceding verb phrase, as we'll see in my example at the end of this post.
Of course the か is merely the sentence-ending question particle here. Thus, putting it all together:
太郎くんは東京大学で勉強したいんじゃないですか。
It would actually probably come out more like
太郎君は東大で勉強したいんじゃないっすか。
Taro, you want to study at Tokyo University, right?
Think of it as "Isn't it true that you want to study at Tokyo University, Taro?" or "Isn't the statement 'Taro wants to study at Tokyo University' true?"