Lets say I send some bitcoin to an address in a normal bitcoin transaction. How many confirmations will it need before another transaction which spends those bitcoins will be accepted by nodes?
Zero.
It is possible (though ill-advised) to send a transaction, and then immediately use that new unspent output to fund another transaction. In general, all nodes will accept and relay both transactions, and all miners (and mining pools) will consider confirming both transaction.
There is some risk however in using the outputs of an unconfirmed transaction to fund a transaction. Due to the malleable nature of a bitcoin transaction ID, it is possible that a modified version of the first transaction which changes its hash will become confirmed, thereby making the second transaction invalid. As such, if at all possible, it is advised to wait for at least 1 confirmation on a transaction before spending any of its outputs.