What is the name of this voting system?

You are given a list of candidates. You can vote for as many as you want, and each vote is counted as one vote for that candidate - they are not ranked. For example-

Jack votes for Candidate A and C.
Jim votes for Candidate C.
Jenna votes for Candidate B and D.
Jennifer votes for Candidate A and C.

Under this system, C has three (full) votes, A has two, and B and D have one each. So C wins.

What’s this voting system called?

It’s called “approval voting.” http://en.wikipedia.org/wiki/Approval_voting

That’s called “Approval Voting”. You basically give each candidate a yea or a nay.

That’s the one. Thanks very much.