If I have 20 machines, all equipped with gigabit ethernet NICs, all connected to a gigabit switch, is the maximum volume of data going through the switch per second equal to 1Gbps, or is it more like 1Gbps * 20?
It is closer to 1Gbps * 20.
Throughput is usually written in the documentation and is expressed in packets per second, which can be translated to mega/gigabits per second depending on packet sizes.
It really depends on the switch, its intended use, price, etc.
For example, cheap SOHO 8 port L2 gigabit switches usually get throughputs from 1–5Gbps (on all ports together — out of 16Gbit full-duplex theoretical max). Enterprise/datacenter/ISP switches can do a lot more.