I used bitaddress to generate a paper wallet offline, it gave me a public, and private key. Do I need to do anything else or can I start to send coins to it?
Is the public key that it gave me the receiving address?
Yes - any BTC sent to that address can be redeemed by the holder of the private key (which therefore needs to be securely held with as much attention to the security as you care for the BTC it can unlock).
and the private key is the password?
It's not really a password but is simply the "key" that is required in order to create a transaction to send the funds from the public address to another (to do this typically you first would import that private key into a wallet).
Most importantly:
1) If any person or (more likely) any spying software has seen your private key then it is not secure and the funds can be stolen.
2) Print out the private key and lock it in a safe and/or encrypt the private key (with a secure password created by some software such as "lastpass") and keep backups of it.
3) Do not lose the private key or your way of recovering it (be it by paper, password or better still both) as funds *cannot* be recovered otherwise.