Argentine tango music is complex, varied and fantastically good. The bulk of what you get to hear in the milongas was written in the first half of the last century.
During what is known as the 'Golden Age of tango music', between the mid-thirties and forties of the twentieth century, a great number of talented orchestras recorded so many tunes that it takes years even for the discerning tango nerd to collect them all. When the military took over Argentina in the fifties both tango music and dance went into a long decline. Some of the well established orchestras still existed, but the recordings are often less danceable, probably just documenting how habits had changed. During that time musicians like Astor Piazzolla also tried to push the envelope of traditional tango music by integrating jazz and classical elements, calling this new form 'Tango Nuevo'. Very few of these pieces are intuitively easy to dance to, stunning as they might be musically. Astor Piazzolla even wrote a Tango Opera, called 'Maria de Buenos Aires'.
The development of tango dance music commenced only recently, during the 1990s, when tango as a danceform became wildly popular again, not only in Buenos Aires, but also in the States, Japan and Europe. To distinguish this - often electronic - dance music for discerning young tango dancers from the music Piazzolla had introduced, it was termed 'Neotango'. Nowadays, 'Neotango' is often used to label anything from Buenos Aires lounge music to European schmaltz to slow pieces by Nora Jones or Whitney Houston. Tango afficionados all over the world have been seen to get all hobby-horsical about what one can rightfully dance to and still call ones dance 'Argentine tango'. Ben and Katja are not fussed about definitions and the like...but Katja is known to drop her dancepartner on the spot as soon as some 'Neotango' comes on.
To learn more about tango music, Todo Tango is an excellent resource. You can browse and download lyrics, scores and all sorts of other information, and you can listen to a great many tunes as well. If you want to buy CDs, you can order them from Zival's tango store, a very good CD shop in Buenos Aires with a reliable mail order service, or you can order from Mike Lavocah's webstore. Mike is a UK tango teacher and Dj who offers a lot of advice about the different orchestras on his website.
If you play an instrument and would like to play tango, there is an exciting movement to create tango community orchestras, lovingly nurtured by Korey Ireland. In the UK, Totnes and Manchester have started forming some, and if you're interested, Korey could most probably be convinced to come to Oxford and give us a headstart.
Lastly, consider that if you buy CDs rather than rip them from a friend you support the Argentine economy...