A picket is a vertical wooden board in a fence. You might have a white picket fence enclosing your whole front yard, so your little dog can't escape.
Besides the "wooden stake or strip" meaning, you can also use picket as a verb to mean "protest or strike." When disgruntled workers go on strike, they often picket outside their workplace, holding signs and chanting. You can also call one of these protesting workers a picket. The original meaning, from the 1680s, is "a pointed stake used as a defensive weapon." And the original picket line was a line of military troops.