What is a wallet, exactly (to an implementation level)?

Is a wallet a client? To what server? A file? How does the file get updated? Is the client P2P? How does it know what peer to reach for? Where is the list of “trusted peers” (in that case)? Do I need a third party to have a wallet?

I’m planning to put my money here, as I understood that most of the times, when Bitcoin “goes up” or “goes very down” it’s actually most currencies going down and up respectively, so I don’t fear “volatility” (where it doesn’t exist.)

So I better understand every bit of it :)

Thanks in advance for any advice!