Synaptic facilitation is primarily caused by elevations in pre-synaptic calcium. Synaptic depression can be caused either by pre-synaptic depletion of vesicles or by post-synaptic release of retrograde messengers.
For more, see this excellent review by Zucker and Regehr. It's slightly out-of-date, but good enough for general purposes.
A Preamble on Relevance
Most models of neural circuits are reducible to some form of graph: nodes that communicate along edges.
Graphs are a powerful model for computation, whether its the computational graph used in Google's TensorFlow software, which underlies their "artificial intelligence" products, or the statistical dependency graphs that power Bayesian inference models. Graphs can even make multi-variable calculus easier to understand!
The inspiration for these models goes back to the observations of Ramon y Cajal: neurons are discrete units that communicate with each other in a defined manner, i.e. across synapses electrical and chemical.
Unfortunately, most graph formalisms assume that these connections are static. In some models, these connections are allowed to change slowly, as in long-term potentiation, where "slowly" is defined relative to the changes in the inputs.
Most of these models break down if that separation of time scales is disrespected. Unfortunately for this class of models, which includes the wildly successful artificial neural networks that underlie Apple's Siri and Facebook's face detection, to name just two applications, real neurons have synaptic weights that change rapidly: from spike to spike, weights can go up, go down, or do both!
Modellers need not despair: these "adaptation" mechanisms can sometimes be quite naturally be incorporated, and can even improve the performance of a model.
With that in mind, let's consider the most common mechanisms for this phenomenon, most commonly referred to as "short-term plasticity".
Now might be a good time to review the mechanisms of synaptic transmission if you're unfamiliar with them.
In synaptic facilitation, the post-synaptic response to a second action potential is greater than its response to the first. The most widely-accepted mechanism is build-up of presynaptic intracellular calcium.
Synaptic release is caused by a transient increase in the concentration of the second-messenger calcium, flowing down its electrochemical gradient through voltage-gated calcium channels due to the depolarization caused by an action potential.
This calcium increase is transient because the concentration is buffered, just as acid-base solutions buffer hydronium and hydroxide ions. Buffering mechanisms include storing calcium internally, extruding it into the extracellular fluid, and allowing it to passively diffuse out of the post-synaptic region.
All of these mechanisms take time: they result in an exponential decay in calcium concentrations, rather than an instantaneous one.
If another action potential enters the pre-synaptic area before these mechanisms have had time to work, the resulting spike in calcium will reach a higher level. Even though this addition is sub-linear, it can result in supra-linear or sublinear changes in the post-synaptic response, since the calcium signal is transformed into vesicular release by a cascade of non-linear mechanisms.
In synaptic depression, the post-synaptic response to a second action potential is less than its response to the first.
Synaptic depression trumps facilitation at most synapses, as might be expected from its key role in homeostatic plasticity for recurrent networks.
The most common pre-synaptic mechanism for depression is depletion of the readily-available pool of vesicles. Vesicles, which mediate the extrusion of neurotransmitter into the synapse, are a finite but renewable resource. If the second action potential comes at a time in between the depletion and the renewal of this resource, then the number of vesicles released will be lower (or even 0!), and so the post-synaptic response will be decreased.
Another well-studied mechanism is the release of retrograde signalling molecules, in particular the endocannabinoid 2-arachidonoylglycerol (2-AG). Unlike classical neurotransmitters, these lipid-soluble molecules can cross the cell membrane, meaning that they don't need to be packed into vesicles or detected by cell-surface receptors.
One precise biomolecular mechanism for this pathway involves activation of the G-protein coupled receptor CB1, which leads to the inhibition of voltage-gated calcium channels (see this review by Ivan Soltesz, Bradley Alger, and others).