Corin's covered most of it already. The reason for this aggression is that hierarchy within the group is really important to tiger barbs. If you have a big group you will see them constantly measuring up to each other, having pushing contests and generally trying to establish who is the top dog. Of course if you only have a pair, this means one poor fish, the weaker one, is going to get it in the neck all the time. In this situation fin-nipping and aggression often seems to extend to other species too. The top dog can't stop bullying, it's his instinct to work frantically to keep on top, he can't stop and say to himself, hang on, I don't need to do this. Also, because tanks are small, the inferior fish can't put a stop to the situation by swimming away, so they're stuck in this vicious circle. In a big enough group, the top dog can exercise his needs on different fish of his own species, so the results aren't usually that disastrous.
The other problem is they are very boisterous and active fish, so they need a lot of room; they do tend to get more aggressive if space is confined. Plus it is more difficult for other fish to get away.