The equations of general relativity work just as well run backwards in time. A black hole is a region that anything can fall into and nothing can leave. Run the film in reverse, and you get its opposite: a region that nothing can fall into and that everything must leave. That is a white hole.
In the mathematics
The idea comes from the complete mathematical description of a simple, non-rotating black hole. Written out in full, in the coordinates found by Martin Kruskal and George Szekeres, the solution has four regions: our universe outside, the black hole, a second, parallel universe outside, and a white hole. Matter and light can escape from the white hole region, but can never return to it. The Soviet physicist Igor Novikov discussed such objects in 1964.
Stephen Hawking argued that in a sense black and white holes may be two faces of the same thing: the time reversal of a black hole in thermal equilibrium is a white hole in thermal equilibrium.
Why physicists doubt them
There is no evidence that white holes exist, and very few scientists expect them to. The white hole region in the mathematics exists only for an eternal black hole. Black holes that form from collapsing stars, the only kind we know, do not have one.
They would also be fragile. In 1974 Douglas Eardley showed that a white hole would be unstable: matter falling towards it from outside would pile up and quickly turn it into a black hole. And there is no known way for one to form in the first place.
Bold ideas
Some physicists have explored the idea anyway. In 2011 Alon Retter and Shlomo Heller suggested that a white hole would release all its matter in a single burst, a “Small Bang”, and proposed that some unusual gamma-ray bursts might be white holes. They also noted that the Big Bang itself has been described as a kind of white hole. In 2014 Hal Haggard and Carlo Rovelli proposed that quantum effects might eventually turn a black hole into a white hole, releasing what fell in long ago.
Quasars were once considered as possible white holes. That idea has long been abandoned: they are powered by supermassive black holes. White holes remain a possibility of the mathematics with no known place in nature.