Zeus threw lightning bolts
He is a sky god: people thought of him as living on top of a mountain (Mount Olympus), and when he is angry he throws lightning bolts out of the sky at people. Zeus probably was not worshipped in Greece before the Indo-European Greeks arrived there in the Middle Bronze Age.
He is an Indo-European god, and he is basically the same as other Indo-European sky gods like Jupiter (this is really even the same word: Ju= Zeus and piter means father) or Odin. He's related to Indra, a Hindu sky god, the son of Dyeus Pita.
In some ways Zeus represents the Indo-Europeans in the minds of the later Greeks. When he arrived in Greece, he seems to have married the local earth goddess, Demeter (De for earth and meter means mother). This may be a sort of religious representation or memory of the actual mixing that took place between the invaders and the local people in the Middle Bronze Age. It is also a fertility ritual, where as with Gaia the sky and the earth come together to make the grain grow: and the child of Zeus and Demeter is Persephone, the grain.
In other ways, people thought of Zeus as representing the Father in psychological terms. People who had real problems with their own fathers might think of Zeus as a sort of ideal father, for instance. In the story of Phaethon, we see Zeus playing the part of the wise father.
Other people might see Zeus more as a way of explaining why sometimes good things happen to you and sometimes bad things: if good things happened, they would say, that was because you had done something that pleased Zeus, and if bad things happened you had done something wrong, and you needed to try to make Zeus happy now. The story of Prometheus explains how this works.
To find out more about Zeus, check out these books from Amazon.com or from your local library:
Lord of the Sky: Zeus, by Doris Gates. Stories about Zeus.
D'aulaire's Book of Greek Myths, by Edgar and Ingri D'Aulaire.
Greek Religion, by Walter Burkert (reprinted 1987). By a leading expert, for adults. He has sections on each of the Greek gods, and discusses their deeper meanings, and their function in Greek society.