I'm trying to get involved in bitcoins, but first it seems I need further clarification on the concepts involved in bitcoins.
First let me clarify what I know (at least what I believe to be correct): miners race to solve mathematical problems, and whoever (or whatever pool) manages to solve the most complex block at a given time is awarded the bitcoins to their wallet address.
Now, what I want to know is, what exactly makes a wallet wallet? Other than providing an address, what does it do? A pair of cryptographic keys make up a wallet, which I understand, is used to transact the bitcoins associated with an address; but is that all? Other than safeguarding the private key, is there anything else to mind about a wallet, i.e. does it record anything else?
I did not find anything explanatory aside from the key-pair mention on the Wikipedia article, and I do not have enough technical expertise to go through the bitcoin protocol to understand the concepts. Please help me with understanding wallets.