Who is Dionysos?
For the Greeks, Dionysos was a fertility god like Demeter. But while Demeter was the goddess of dry things growing, like grain, Dionysos was the god of wet things growing, like fruit of all kinds, but especially grapes (and the wine that people made from grapes). In fact, Dionysos is in some ways more like Persephone than like Demeter, because he is often thought of as being the wine itself, just as Persephone is the wheat.
For instance, when people drank wine, they said that they were taking the god into their own bodies, and when they became tipsy, they said that the god had taken over their minds and hearts. The Greek word for this is "enthusiastic", en= in and thus = god, taking the god into you. You were not really responsible for things you did while you were drunk, because the god made you do them.
Mythologically, Dionysos is one of the younger gods, like Apollo, Artemis, and Athena. He is even younger than they are. In fact, he is so young that Homer, in the Iliad and the Odyssey, seems not to know about him yet - although his name does show up in written documents from the Late Bronze Age in Greece (about 1300 BC). He is the son of Zeus, and a mortal women named Semele (SEH-muh-lay). The story goes that Zeus and Semele were in love, and she was going to have his baby, and he was so happy about it that he told her (foolishly) that he would give her whatever she wished for, anything at all (in fairy tales, and maybe in real life, this is ALWAYS a bad idea! - compare for instance the Christian story of Salome).
Semele wished to see Zeus in his true form, but Zeus' true form is that of a lightning bolt! That would kill her! Zeus tried to talk her out of it, but she insisted. So he had to do it. But when he appeared right next to her as a lightning bolt, of course she was killed. Zeus managed to rescue their unborn child and sewed the baby up in his own leg until it was born, and that was the baby Dionysos.
Because Dionysos is the son of Zeus, and the Greeks thought of him as foreign, and the Hindu god Indra is the son of Dyeus Pita, Dionysos may have started out as one form of the Indian god Indra, but there isn't much evidence either way, and the earliest references we have to Indra are from a little later, about 1000 BC. Dionysos might also be related to the German god Odin, also a younger god, and also a god who takes over people's minds and makes them lose control of themselves.
Because wine makes people lose control of themselves, a lot of stories about Dionysos have to do with the Greek idea of sophrosyne, or self-control. As you can see in the story of Phaedra, the Greeks thought it was good to be able to control your emotions, but not all the time. People should also be able to let themselves go enough to fall in love, or enough to just relax and enjoy themselves sometimes. The story of the Bacchae is Euripides' version of this idea.
Learn by doing: Talk to somebody about what it's like to be drunk
More about Dionysos
The Spirit of Spring: A Tale of the Greek God Dionysos, by Penelope Proddow (1994). Easy reading.
D'aulaire's Book of Greek Myths, by Edgar and Ingri D'Aulaire.
Greek Religion, by Walter Burkert (reprinted 1987). By a leading expert. He has sections on each of the Greek gods, and discusses their deeper meanings, and their function in Greek society.
Ancient Mystery Cults, by Walter Burkert (reprinted 1989). More about Dionysos and other mystery cults.