Plurality
The usual way to vote: everyone picks one option, and the one with the most votes wins, even without a majority. It's also called first-past-the-post, and most elections in the English-speaking world use it. Everyone knows how it works; the trouble starts when two options are alike.
The problem: vote-splitting
When two similar candidates run, they split the votes of everyone who likes them both, and a third candidate most voters like less can win. So people learn to vote for someone who can win rather than the one they like best. It's why countries that vote this way tend to end up with two big parties (Duverger's law), and why so many people vote for "the lesser of two evils."
In the real world
In the 2000 US presidential election, Ralph Nader received 97,000 votes in Florida. George W. Bush won the state by 537. Whether Nader "cost" Gore the election is still argued over, but it's the example people reach for when they explain vote-splitting.
When to use it
Plurality is fine for two options, or when one is a clear favorite. With three or more, similar options split the vote, so TallyLab suggests approval for most group decisions.