To snigger is to laugh scornfully, especially when you're trying to hide the fact that you're laughing. Kids might snigger when the biggest show-off in class falls over backward in his seat.
You can generally use the verbs snigger and snicker interchangeably. They both mean "to snort with partially suppressed laughter," and both imply a sense of superiority or scorn. You're most likely to snigger at someone when they're being ridiculous or foolish. Experts guess that both versions of this word are imitative in origin.