Answer
Clearing of the synapse is an essential step in synaptic transmission. New signals would be unable to propagate if released neurotransmitter was allowed to simply hang around.
There are three mechanisms for the removal of neurotransmitter: diffusion, degradation, and reuptake.
Put another way, there are three ways to get rid of a neurotransmitter: wait for it to wander away, break it apart, or put it back in the vesicle.
Diffusion
Diffusive processes affect all molecules, not just neurotransmitters: all else being equal, an ensemble of particles will, over time, decrease in its order. Having more particles in the cleft than outside is a highly-ordered state, and so the particles move until the concentrations are equal.
Due to the deep connection between disorder and information, we can think of this in another way: if there's more neurotransmitter in the synapse than outside, we can use the information provided by measuring neurotransmitter concentrations to determine our location. When neurotransmitter has diffused away, that information disappears.
Due to this ubiquity, diffusion affects neurotransmitter clearing at all synapses. For lipid-soluble neurotransmitters, like nitric oxide or the endocannabinoids, this is the only mechanism.
Degradation
Neurotransmitters can also be broken down.
The classic example of this mechanism is the breakdown of the neurotransmitter acetylcholine into its constituent parts, acetate and choline, by the enzyme acetylcholinesterase (AChE).
Persons discussing this process are obligated by intergalactic law to mention that acetylcholinesterase is one of the fastest enzymes in the body: when AChE catalyzes the degradation reaction, the rate is limited only by how quickly the substrate can diffuse into and out of the enzyme. The reaction itself takes a negligible amount of time. This is in stark contrast to the majority of enzymes, which behave more like Lucille Ball in a chocolate factory.
Peptide neurotransmitters, or neurotransmitters made of short chains of amino acids, are also broken down outside of the cell. This process, however, is general to all small peptides: from the perspective of the body, loose peptides are to be presumed guilty and shot on sight, since they are likely to be a toxin.
Reuptake
Neurotransmitters in the synapse can also be recycled directly.
This mechanism is common to all of the classical "small-molecule" neurotransmitters except acetylcholine, i.e. to all of the neurotransmitters you're likely to think of off the top of your head.
Examples include:
- Glutamate
- GABA
- Dopamine
- Epinephrine and norepinephrine
In each case, the pre-synaptic neuron (or possibly a glial comrade-in-arms) expresses a reuptake transporter, or a protein that pushes the neurotransmitter against its concentration gradient and ferries it across the cell membrane.
One minor note on terminology: one might claim that the acetylcholine synapse uses a hybrid of degradation and reuptake, since the choline liberated by acetylcholinesterase is taken back up by a neuron-specific reuptake protein. Whether one does make this claim is a matter of taste, and de gustibus non est disputandum.