The following is a big newbie question, I am aware. Don't judge me, k?
If I never mined bitcoins, and never installed a bitcoin client, but I *did* sign up on of the exchanges (say mtgox), do I still have a wallet? Or are the bitcoins I buy on an exchange stored in some kind of "proprietary" wallet of the exchange I signed up for?
The bitcoins are in a "proprietary" wallet of the exchange you signed up for. You are trusting them to hold on to those bitcoins and keep them safe for you. You are trusting them to send those bitcoins in a timely manner to any address you ask them to in the future. You are trusting them to still be there in the future when you want to access those bitcoins.
Here are some commonly used wallets, you can research each on your own time and determine which best suits your needs:
Paper Wallet
Armory
Bitcoin-Qt
Electrum
MultiBit
https://blockchain.info/walletAhh, thank you for the response. So a wallet is basically just a way of keeping track of the addresses - those addresses are essentially in no way tied together - other than who 'owns' them. Would that be a correct statement?
Yes, that would be a correct statement.
Santoshi's Satoshi's article states "We define an electronic coin as a chain of digital signatures.". How does that chain of digital signatures get stored in a wallet?
The chain (of all digital signatures associated with all bitcoins) is stored in the blockchain. The blockchain is a verifiable digital ledger that every full peer maintains a complete copy of. When miners create a new block, they broadcast it and all peers add the block to their own copy of the blockchain after validating it.
Each time a coin (or fraction thereof) a new block is added to chain. Does that mean the digital signatures get longer and longer?
Figuratively I suppose you could look at it that way. In reality the blockchain gets longer and longer. Each new signature just has a reference back to the location of the previous signature in the blockchain file(s).
A block in the chain contains the addresses of both parties. Are those public addresses which correspond to private addresses?
There are not private addresses. There are private keys. All addresses are considered public (although the public may not know about them until you receive bitcoins at them). A transaction has a reference to the public bitcoin address of the receiving party. Since the transaction also has a reference to the previous transaction where the sending party was the receiving party, it is possible to find the bitcoin addresses that were used by the sending party to receive the bitcoins. The private keys are used by the wallet software to generate the digital signature required to sign over control of the value to the new address. The private keys used for the signature are the private keys that were used to generate the bitcoin addresses where the sending party received the value that they are sending.
Are those private addresses the privkeys you refer to?
The private keys that the wallet software maintains when it generates a new bitcoin address, and uses to create digital signatures to sign over control of value received at those addresses to someone else's address, are the privkeys being referred to.
Do I have privkeys for every bitcoin ( or fraction thereof) I receive?
That depends. If you use a new address for every transaction, then yes, you have a private key for every bitcoin (or fraction or multiple thereof) that you receive. If on the other hand you re-use addresses multiple times, then you have less private keys. Specifically, you have a private key for every bitcoin address that your wallet is keeping track of.
Note that, depending on the wallet software you choose to use, it is possible that your wallet has created additional addresses that it hasn't told you about and sent some of your bitcoins to those addresses as well. This would be true of any wallet (such as Bitcoin-Qt) that generates a new address for the "change" in every transaction sent.
Should I consolidate them by doing a transaction to myself.
If you are receiving lots of transactions that are all less than 0.5 BTC (such as players of SatoshiDice encounter), then you might want to consider consolidating the outputs received before you end up with hundreds of outputs all valued at less than 0.01 BTC.
I apologize for so many questions. Thanks for any replies.
No need to apologize. Asking about the things you don't understand is a good way to learn. This is the Newbie forum after all.