Spike Timing Dependent Plasticity and Hebbian Learning

Hebb (in 1949) said that if neuron A tends to help neuron B to fire, it will become better at doing it over time.
There is a lot of biological evidence that this happens, and the evidence continues to grow.
Perhaps the best supported, most investigated, and most used Hebbian mechanism is spike timing dependent plasticity (STDP).
If neuron A fires a few milliseconds before B, and there is already a synapse from A to B, the weight increases. If B fires before A, the weight decreases.
Note the assumptions that the weight is positive, and that both have fired.
It's often used so that the weight goes up if A and B fire at the same time, though I think the biological evidence here is weak.
The weight goes up more (or down more) the closer the time difference is to 0.
Let me be quite explicit this is nothing like a complete mechanism for learning in the brain.
It is widely available in the standard simulators, and there are several versions in NEST (and on SpiNNaker).