When someone doesn't seem to care about the misfortunes of others, they're unsympathetic. If you yawn while your friend is crying about her lost cat, you'll come across as terribly unsympathetic.
There are a few ways to use unsympathetic. If you lack sympathy (a feeling of sadness on behalf of someone else), you're unsympathetic: "I'm unsympathetic about her missing cashmere scarf, since I happen to know she has three more at home." It can also mean "unappealing" or "unlikeable." Darth Vader, up to a point, is a decidedly unsympathetic movie character. Finally, it can mean "disapproving," as in your unsympathetic opinion of an elected official whose policies you disagree with.