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.

The term canonical comes from the concept of canon, that is, when you follow a standardized way to do something, you follow the canon, hence the thing that you do is canonical.

In general, a canon is defined when something can be constructed in many different ways and we choose one because it is more convenient. By example there are canonical (standard) cut branches of many commons complex functions.

Also the concept canonical is very similar to the concept conventional

IMO the best place to comprehend this concept is when you see canonical forms applied to matrices, graphs or word problems to set representatives for each equivalent class of objects: you need to define a convenient algebraic manipulation to define these representatives uniquely. Then you can work over these canonical representatives instead of the whole classes, this can simplify the proof of some theorems.

This wikipedia article about canonicalization in computer science is analogous (or equivalent) to the canonicalization in mathemathics.