The short answer
About 13.8 billion years ago, all the matter and energy in the observable universe was compressed into an extraordinarily hot, dense state. Space began to expand — rapidly at first, then more slowly — and has been expanding ever since. This is the Big Bang.
What it was not
The biggest misconception: the Big Bang was not an explosion of matter into empty pre-existing space. There was no empty space to explode into. Space itself came into existence and expanded. There is no "centre" of the explosion, because every point in the universe was the starting point. Every galaxy sees every other galaxy moving away from it — not because galaxies are moving through space, but because the space between them is growing. This is cosmological redshift, first observed by Edwin Hubble in 1929.
Three pieces of evidence
Distant galaxies are moving away from us — and the further they are, the faster they recede. This relationship (Hubble's Law) implies the universe was once much smaller and denser. HubbleSite explains Hubble's Law.
The afterglow of the hot early universe still fills the sky as faint microwave radiation at 2.725 K. Predicted in 1948, accidentally discovered in 1965 by Penzias and Wilson. Its tiny temperature fluctuations match the predicted seeds of today's galaxy clusters to extraordinary precision.
In the first three minutes, the universe was hot enough for nuclear fusion. The theory predicts specific ratios of hydrogen (75%), helium (25%) and traces of lithium. Observations of the oldest stars match these predictions closely — which would be an extraordinary coincidence if the Big Bang had not occurred.