It's supposed to be bad luck
to sit at a table at which there are exactly thirteen
people. The result is either general bad luck, or the specific prediction
that one of the thirteen will die within a year, or even more specifically that the first to leave the table will be the first to die.
This is part of a whole series of superstitions about the number 13 (see also: triskaidekaphobia), but has a couple of specific possible origins.
Norse mythology has it that there was a banquet held at Valhalla, to which twelve gods were invited. Loki, the god of strife, chaos or deceit, arrived, and killed Balder, the god of light and beauty.
The Christianised explanation is that 13 is unlucky because there were 13 people at The Last Supper, and the 13th was either Judas, who betrayed Jesus, or Jesus himself, who was betrayed shortly afterwards. It's also Judas who was said to be first to leave the Last Supper, which is an illogical relationship to the 'first to leave, first to die' belief, as it wasn't he who died. However, it's possible that the Christian aversion to the number 13 actually came about because the number 13 is sacred in some Pagan religions, including Wicca.
The bad luck is lessened somewhat if all 13 people stand up together to leave. Because, presumably, that means they'll all die, simultaneously, within a year. How comforting.