Baixa is the historical old town of Lisbon. It is one of the most beautiful and visited neighbourhoods in the whole city. It is very, very popular among the tourists, travellers and backpackers visiting Lisbon.
Living in Baixa is living in the heart of the city. This neighbourhood is famous for its perfectly rectangular blocks, its 18th Century architecture and the beautiful "Portuguese pavements" (google it!).
Living in Baixa is ridiculously convenient: this is the very centre of the city after all. You have metro stations in every direction (Baixa-Chiado, Rossio, Terreiro do Paço), plenty of bus-stops, hundreds of shops, quite a few markets and all the restaurants you could wish for.