I started playing concertina just this year. In general, I'm learning for my one year old son, because we want to raise him in a musical household. My husband has played violin (and several other instruments) since childhood, and until the end of my schooling days I played piano and organ. I wanted to get back into playing music, but with something more portable and compatible with the violin. I'd tried electronic keyboard a few times, but while it was fun, it never stuck. I think I wanted something analog because most of the rest of my life is in the digital world. I'm not sure how I came across the concertina, but I knew that's what I was looking for as soon as I saw it.
I'm in the same camp as Matthew and Sarah - prior to finding players on YouTube I think my only exposure was incidentially in movies, but I have heard a lot of accordion music, mostly folk and pop. So far I play American folk, children's songs, parts of the beginner Suzuki repertoire, and a smattering of traditional European music.