A fancy word for "place where someone is performing". It can be anything from a Starbucks to a nightclub to a rented community center, whatever. The performer can be a band, a comedian, a middle school theater club, or anything else that involves performing. If somebody is performing somewhere, that somewhere is the venue.
Jake - "Hey, I heard Brad's band is playing at Red Column House tomorrow night"
Lake - "Yeah? That's a good venue. Sure beats the armpit they were in last week"
Comes from the term "gig". It means you are moving from one gig to the next, singing along with each band for a few songs and then moving on. It's similar to clubbing, except the clubs are more likely to be coffee shops and there will probably be more singing and less dancing/drinking. Gigging is about experiencing the music and atmosphere of each venue. Tends to be more popular with people who enjoy jazz or "underground" music.
Me (PE major) - "Hey bro, want to shoot some hoops with Derek later?"
Friend (Jazz Guitar major) "Naw, I'm going gigging with some guys from my Theory 2 class."