What is the term for learning to group objects I've been googling this for a few days now but haven't come across what I'm looking for. 

Scenario: A child has to learn how to identify species. When first learning, they might mistake a cat for a dog. They both have four legs and a tail. However, as they are shown more and more examples of cats and dogs, they build up a more detailed definition of what a dog is and what a cat is ... specifically by being shown many many examples (different examples at all ends of the spectrum ... big and small) of these two species. 

I was reading about this area of cognitive psychology a few years back (quite a few years back). I wanted to refresh and study up on this some and read what I can about it, but can't because I'm not googling the correct terminology. Any direction would be greatly appreciated. Thanks.
I would suggest you start with the Wikipedia articles on Categorization and more specifically Prototype theory.

While the Categorization article gives a number of branching points for you to drill down on and hone in and/or prime your thinking for further searches, the Prototype theory topic is where my mind went upon first reading your question

Even though the focus of prototype theory is on the shared artifact of language which you may or may not have been thinking of when you posed the question. I would expect any further reading you did on the topic to have plenty to say about the individual language user/learner also. 

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As a follow up I also found the following article which is from the perspective of Developmental Psychology.