Isaiah usually brings up the potter and clay image when God is making a tough point about his people's disobedience, stating that they can't refute him or go against his will any more than a clay pot can rebel against the potter who makes it. God is the Maker, his people are the Made. As he says in Isaiah 29:16:
You turn things upside down! Shall the potter be regarded as the clay? Shall the thing made say of its maker, 'He did not make me'; or the thing formed say of the one who formed it, 'He has no understanding'?
Also, the potter and clay imagery helps to explain what God is doing to his people when he bombards them with his wrath. In some sense, he's like a potter smashing a clay pot he already made, melting it back into clay, and then remaking the pot from the clay. It's a way of purging and refining Israel, removing its imperfections and forming it into a more perfect whole.