The weak nuclear force (or just the weak force, or weak interaction) acts inside individual nucleons, which means that it is even shorter ranged than the strong force. It is the force that allows protons to turn into neutrons and vice versa through beta decay. See size of the universe for a list of visuals about how short ranged the weak force is. This lets a nucleus keep the right balance of protons and neutrons to keep everything together. Without the weak force, fusion, in the sun or on Earth, would be impossible.
As the name implies, the weak force is much weaker than the strong force, or the electromagnetic force, but it's still quite a bit stronger than the gravitational force.
Modern physics has unified the electromagnetic and weak forces into the electroweak force. People continue to try to unify all of the forces in a grand unified theory.
Fully understanding the weak force takes many years of study, but some fun places to start include hyperphysics or the blog of Prof. Matt Strassler.
Below is the Scishow's series on fundamental forces part 2, the weak force:
The remaining videos can be found on the pages for gravity, strong nuclear force, and electromagnetic force.