When you dodder, you walk in a shaky or trembling way. You're most likely to see older people dodder, moving slowing and unsteadily.
People usually dodder because of illness, disability, or the fragility of very old age. Your great Aunt Martha may dodder over to you, but then give you a powerful bear hug when she gets there. The verb dodder dates from the 1600's, and it's thought to have probably come from the Middle English daderen, "to quake or tremble."