The salsa club is called Randall's, and it's located near the Moncloa/Plaza de España area, which happens to be very close to where I live.
Typically they have a lesson at 11pm for an hour, then two hours of open dancing until 2am, another mini-lesson at 2am, and then open dancing for a while more.
The reason I thought this salsa club/bar was awesome was because it wasn't super crowded on a weekday night. This means that the only people there were locals! Places with lots of tourists are crowded always at all times of day, but because this wasn't super crowded, I knew it was authentic.
Salsa is not a Spanish dance; it is Latin American in origin. But there are plenty of Latin American immigrants who live in Spain (note, one of my flatmates is such a one) and have lived there for many years, so they are as much "locals" as half the population of the United States (blog post about what it means to be a local coming soon...). In any case, dancing with Jess's co-workers was fun; they danced salsa very well, since they were all mostly Latin American. I did not dance with anyone who was not from our original group, mainly because they were much much better than we were and didn't need to ask us to dance.
As a girl dancing salsa in both the US and Spain, all you need to do is wait for the guy to ask you to dance; make eye contact and know you're interested in dancing, but the rest is up to him. I definitely want to go back at some point to catch the lesson and improve my salsa skillz!