To tat is to make lace. Tatting involves making loops and tying knots in a long piece of thread over and over again.
Most of the lace made today is created with a machine, and the ability to tat, or make lace by hand, is rare. Tatting requires a lot of silk or cotton thread and either a metal disc called a "shuttle," needles, or a crochet hook. You may also find tat used as a noun to mean "cheap, tasteless stuff," or informally to mean "tattoo," as in "Hey, I love the new tat on your shoulder."