When you unfurl something, you unroll it or spread it out. Your yoga teacher will unfurl her long purple mat at the beginning of class.
To unfurl something is to roll it out, unwinding it the way you do with a coiled laptop charger cord. You might also unfurl a pair of socks curled around each other or the American flag you keep rolled up in a closet until the Fourth of July. Unfurl is a more common word than its opposite, furl, which means "roll or fold." The French origin is ferler, "to furl," from the Old French roots fer, "firm" and lier, "bind."