Though David's answer is technically correct, I offer an alternative for those who struggle with the term in broader contexts.
Canonical simply means relating to a historically established paradigm, or relating to a common or standard model (which simplifies things!). For example, a biochemical pathway or a mechanism, for example, may be understood to include certain typical players or long-understood elements or interactions. These are canonical elements. They relate to the scientific canon. They depend on the history of discovery and organization of our theories, rather than any biological phenomenon.
For instance, a synapse is canonically composed of a pre-synaptic neuron and a post-synaptic neuron. This has been long known and to this day scientists broadly speak in terms of the two, as a simplification of the system. However, most synapses are tripartite, including a glial cell cradling and modulating the environment between the communicating neurons. The glial cell can be considered to be the non-canonical element, even though it may be essential, or very important in the system. But it hasn't been studied as long, and introduces more considerations and complexities to think about which go beyond a simple way of thinking about it.
Some pathways or mechanisms have been strictly defined and outlined, where one can clearly draw the line between the canonical and non-canonical elements. There is no flexibility in how you use the word. On the other hand, other times the word is used in cases of systems where there is no strict definition dividing the two, it's just a judgment call, of what most scientists consider central, primary and emblematic (as opposed to peripheral, or secondary, or not validated enough 'to have entered the canon') to the system. There the use of the term canonical is colloquial!
Sometimes it just means "official" or "standardized" or "really important", but usually it has the more precise meaning "relating to the Hamiltonian formulation of classical mechanics". The canonical momenta are usually first introduced in the Lagrangian framework, but they are the momenta that appear in the phase space of Hamiltonian mechanics. Canonical transformations are symmetries of that phase space that preserve the symplectic structure. Canonical perturbation theory is formulated within Hamiltonian mechanics. The canonical commutation relations are a quantized version of Poisson brackets (as per Dirac's quantization rule).
"Canonical" is an informal term often used in mathematics. Sometimes it means you and your neighbour would come up with the same map. Sometimes it means it doesn't use any choice. Sometimes it means it does use some choice but it is independent of such choice.
Suppose we have a mathematical object. There can be many ways of representing an object that are equivalent to this object for the purposes of solving some problem.
Rather than solve a given problem for all possible objects, we often only need to solve the problem for one representative from each equivalence class. Representatives from these equivalence classes can be called canonical; and it is sufficient to solve the problem only for canonical representatives.
We usually choose canonical representatives that are easy for us to work with.
For example, for graphs
We might even allow equivalence classes to have more than one canonical representative. Solving the problem for all canonical representatives nevertheless still amounts to solving the problem for all objects.
As another example, consider Latin squares
However, in the Latin square case, there are usually many ways to permute the rows and columns (and symbols) to get the first row and first column in order. So there would be many reduced (or canonical) representatives from each equivalence class.
