Well, doesn't tango have a lot of cliches to live up to! Tango sits apart from the other ballroom dances, and has a very different history. While foxtrot, waltz, quickstep, and Viennese waltz have their roots firmly in European and Anglo-American culture, tango's roots lie in Argentina. The dance has had many incarnations. From its Argentinian origins, it travelled to North America and changed a bit. Following its popularity there, it became a rage in Britain and Europe, most notably in France, where it evolved further. The new version of tango travelled back to North America and developed further still, forming the basis of what we presently call ballroom tango.
Meanwhile, the dance continued to evolve in a very different direction in its original home of Argentina, and was exported once again more recently as Argentine tango.
We teach both ballroom tango and Argentine tango. The two dances are quite different from each other (although their shared history is evident if you're familiar with both styles), and they're both completely different from the earlier dances which are their shared ancestors.
Ballroom tango is characterised by strong, sharp, dramatic movement, with a slightly aggresive mood. Argentine tango is far softer and more romantic, but nonetheless has elements of tension. If you liken the two dances to relationships (and we've all seen these in action!), ballroom tango is a passionate relationship fraught with argumentative tension, whereas Argentine tango is a tense, uneasy relationship fraught with passion. Is that too cheesy an analogy? Probably, but it's a pretty accurate description.
Tango music is a genre of its own. Here are some examples, some of which have the fire of ballroom tango, while others have a smoothness more suited to the Argentine style. There are some traditional and some contemporary ones here.