"Bumming" is when someone, who otherwise is a "traditional" member (think 1 weekend/month, 2weeks/year (plus a few extra days per month for flying currency)), flies extra. Typically this is because that member is between jobs in their civilian capacity.
So, they are not a full time member, per se, but they might seek full time orders (for a few months to a few years) or, depending on the mission of the unit, they might just fly and take as many trips as they can get their hands on.
In extreme cases, on a day that they weren't even scheduled to fly (or get paid), they might be caught hanging around the ops desk, hawking the fight for potential fallout and flying opportunities. In general terms, being a "bum".