Does my address change if I am on different computers

[testmonkey]

I just downloaded bitcoin on my laptop and I was wondering if I install it on my desktop (the computer that I will be mining on) whether I get to keep the same address and if not how do I make sure I get the same address when I get it on my desktop.

Also when I am mining do I need to keep my bitcoin wallet open or can it be closed.

[lysr]

You only need to keep your wallet(a file containing your public key and private key), to receive payment, you only need to give out your public key, remember never give out your private key to anyone.

No, you don't need to keep your wallet or your bitcoin client opened during mining.

[tysat]

I'd suggest checking out the readme and other links posted up top.  Sounds like you don't know much about bitcoin yet.

[testmonkey]

Ok,

I have bitcoin-qt and when I click receive coins I see an address, is that my public key or private key, if it isn't my private key then were can I see that. Also I have an account on mt. gox and I have a different key there, so should I be using the bitcoin-qt address or the mt.gox one?

[niko]

Ok,

I have bitcoin-qt and when I click receive coins I see an address, is that my public key or private key, if it isn't my private key then were can I see that. Also I have an account on mt. gox and I have a different key there, so should I be using the bitcoin-qt address or the mt.gox one?

Welcome to Bitcoinland! Before you start moving money around, I highly recommend spending a day or two reading about the basics of how Bitcoin operates. This will make us all happier.
You can start on the Bitcoin Wiki, stickies in this board, or redd.it/18kt6y

Here is the broad picture:
Private key is a private, random string.
From Priv Key, public key is derived using a one-way function. Practically impossible to do the reverse if you know a public key.
From public key, the address is derived using a different kind of one-way function. Again, impossible to reverse.

You can generate as many key pairs and addresses as you wish. In fact you should have a new one for each transaction. Most wallet programs and online services do this for you.

Happy reading, and come back here soon!

[Bakemono]

You can use Electrum as a wallet which makes it easy having the same wallet (same collection of addresses/private keys) in different computers.

The address you see in the receive coins tab is like an email address.It is where people are going to send you bitcoins.The private key for that address is stored in your wallet.dat file.(You dont need to know your private key except for backup/restore reasons)

Also if you are mining through a pool (except p2pool) you don't need your bitcoin client running.