You are right about the nonce and the hashing, but..
Everybody is working on a different block.
To pay the block creator 50BTC you need to know the Bitcoin Address it needs to go to.
For the block creator to be able to spend the 50BTC they create, they need to have the private key associated with the bitcoin address the 50 was paid to.
Since everybody has a different private key (randomly generated), everyone has a different Bitcoin Address (associated public key (hashed))
Part of what makes up the block is the hash of the transaction that pays the 50BTC to the block creators Bitcoin Address.
This means that everybody has a different block.
Whoever solves their block first 'wins' and that block is acclaimed by all as the 'next' block.
Oh, so it's both the luck of the draw and a sequential order? Meaning, my computer is trying to solve it's block. It might do it in 1 hour or 100 hours. Once it does, it moves on to another random block?
So an old PC joins the Bit Coin network, works on a block for 4 days and solves it, then it broadcast that out to the network where it's verified before being added to the chain? Does that mean during that 4 days, there is no way someone else is going to solve the same block elsewhere and claim ownership of it?