Some years ago Tom Leinster wrote a wonderful explanation of what makes a (mathematical) definition 'natural':

It’s a matter of aesthetics, and obviously there’s no precise answer, but here’s my stab at it. When you meet a new definition, there are two questions you might ask:

• where does it come from?

and

• why is the definition exactly that, not something slightly different?

If there’s a satisfactory answer to both, you can call the definition ‘natural’.

I've not been as careful about my definitions in the same way but I've been very careful about the notions and theories that I tend to study. To me Tom's explanation expands to these too. E.g. where does a theory (such as group theory, graph theory, ...) originate and why does it look the way it looks?