Balsam is a kind of 'catch-all' word to describe natural resins that are thick, gummy, and or oily. So, in certain contexts, you could think of balsam as meaning 'botanical resin'. Probably the most famous historic balsam would be Balsam Poplars, better known as Balm of Gilead.
I was just sniffing some Balsam Peru (Myroxylon) at a natural chemists shop over the weekend. Balsam Peru is a very, very, VERY lovely scent that is something like vanilla mixed with woods, an extremely tiny note of camphor, and a bit green. It's been described as having a slightly olive scent, which I can detect when it has dried down a little.