Hinduism and Buddhism both emerged from India out of the same Vedic traditions, splitting off from one another around 2,500 years ago.
The oldest texts that could be considered part of the Hindu tradition pre-date the earliest Buddhist texts (and the life of the Buddha himself), but the practice of Buddhism pre-dates the organized Hindu religion that we see today.
Central to Hinduism is a rich tradition of metaphysics and cosmological structure; everything is god, god is everything, and everything that exists is an incarnation of this godhead. God is synonymous with the cosmos, and the self is synonymous with god. In this view, the goal of a Hindu is to live in accordance with Dharma—a sort of cosmic order achieved through unity of the self with the supreme self/consciousness (Atman/Brahman/god).
This is where Buddhism is diametrically opposed to Hinduism: the Buddha taught that there is no god, no soul, no self, that our minds are detached from the world, and that suffering is an innate aspect of human life. For a Buddhist, the ultimate goal is to extinguish suffering and achieve Nirvana (which can be interpreted as everlasting happiness, liberation, and/or emptiness).
This all may seem a little bit confusing at first glance. To paint a simpler picture, the classical Buddhist text, The Dhammapada, begins with these lines:
Whereas Hinduism puts forth the idea that our self is identical to the cosmos, Buddhism flips this on its head, and says that the outer world we perceive as the cosmos is actually contained entirely inside of our own minds.
This idea mirrors that of the Existentialists: that we can never observe an objective reality, nor can we know truths, nor can we see things how they really are—rather, we can only see things the way they appear to us. In this sense, everything we know about the world and the cosmos is contained within our own minds—including how we choose to live our lives.
But because we can't peer into the minds of others, all of our knowledge about the world exists in a sort of mental vacuum, and can only be communicated imperfectly through words (which are also subject to interpretation). This means that we can never really know anything outside of the contents of our own minds.
But all is not lost. One sort of thing that we're able to understand with near certainty is that change and suffering are intrinsically connected:
Human minds are hard-wired to become attached to things: to ideas, to possessions, to livelihoods, to other people, to expectations, and so on. This attachment leads to suffering because nothing is permanent—everything decays, everything dies, and new things take their place.
When we become attached to something, we're bound to eventually lose it, and that loss (or even the expectation of loss) leads us towards despair. The Buddha emphasized that, if we could extinguish our attachment to these transitory things, then we could avoid the suffering caused by their loss.
To accomplish this, Buddhism emphasizes practicing detachment—not necessarily from the world, or from emotions in general, but rather from our subjective experience itself.
Buddhist meditation involves emptying the mind and becoming an observer to one's own mental states and thought processes. By becoming an observer of to our own mind, we're better able to direct our mind towards achieving desirable states—such as happiness, bliss, or liberation—and, most crucially, remain detached from the temporal and fleeting nature of existence.
Those are just some of the things that the Buddha (allegedly) said.