is the 'public/private key' the same as the alpha numerical address to where i sent the BTC?
So can I install any new client and input my public/private key and then my bitcoin will appear?
The address you send the BTCs to is the public "key". It's public so obviously it's not the one giving you access to the Coins.
The private key is another longer set of characters and this is really a key to your public address.
If you input this in any wallet, the Bitcoins will appear.
This. Basically an analogy that I've heard is to consider it like this:
The public key is like a see-through safe. Anyone can see what's inside, but nobody can enter without the actual key.
The private key is what opens the safe. This allows you to not only see what's inside, but also remove its contents.
The private key should never be known to anyone but you. This is stored in your wallet.dat file and it is IMPORTANT that you keep backups of this. Losing your private key means the coins stored on it are lost forever.