The basic theory is this:
Life is suffering. You can never really achieve peace. Happiness is just the flip side of sadness and no material or emotional pleasure will sustain the longing in your soul to understand itself.
The only way to eliminate suffering is to follow Buddha’s precepts to sever your attachments to the world. This doesn’t mean that you cut yourself off but rather that you do all that you do without expectation or desire. You simply do.
Little by little the ego will then slowly fade away until one day you achieve liberation from all selfish motivations and you reach nirvana. You can either opt out of the cycle of rebirth or else stay on to spread the word. The latter are called Boddhisattvas.
The idea that you have God inside you and that it’s simply masked by an ego is basically a Hindu idea. In fact it’s so hard to separate the two that some Hindus claim (quite typically) that Buddha was simply another incarnation of Vishnu, a god who certainly gets about.
Buddhism is quite distinct from Hinduism in practice but in theory they can seem confusingly similar. Both hold that it is our ego that prevents us from seeing what we really are – a part of God. Like a drop of water separated from the ocean and longing to return. But in fact mystics everywhere talk about that and use the same imagery.
Both believe in reincarnation and that what you do in this life, the sum of all your karma, will influence your rebirth in the next. Both see the everyday world as maya, the never-ending dance of illusion. The scholars argue that there are important differences in how the two religions understand the soul and it’s union with Brahman or Nirvana. It seems to be largely a question of semantics though.
However on the social level Buddhism is far more enlightened as it recognises no difference in social castes and holds that anyone can achieve Nirvana whether he be a king or a beggar. In this sense Buddhism was quite revolutionary in India and thus not so popular with the Brahmins interested in maintaining the status quo.
After Buddha lived a number of years as an ascetic he decided that it was better to follow a middle way. A path somewhere between enjoying the things of this world and living in utter austerity. Hinduism on the other hand celebrates the renunciate who turns his back on the world and lives say, on a couple of bananas a day. So perhaps, really, the differences are as much a question of attitude as anything else.