In plain approval voting, you either approve of a candidate or not, and you can approve of more than one. The most approvals wins the race. In net approval voting, you vote up, down, or neutral on each candidate. The candidate with the most approvals minus specific disapprovals wins.
Net approval actually works with candidates much like discussion sites with both up and down votes for posts treat those posts. A controversial candidate may get many approvals and many disapprovals. A fairly popular candidate who's not as divisive might have some fewer approvals but far fewer disapprovals. It should feel natural for people accustomed to reddit, HN, similar sites, or call center feedback systems (where often 9 and 10 of 10 are positive scores, 6-8 are neutral, and 1-5 are actually considered negative).