When you make a transaction it is added to the network queue. At that point miners have already taken transactions from the queue, added it to their block and started mining for a correct hash.  Once a hast has been found, the winning block propagated to the network, miners take the hash of the just completed block, add new transactions to their block and start mining again. Depending upon how backlogged the network is how long your transaction will be unconfirmed.

Waiting thirty minutes means that 3 blocks have passed between the time you pressed sent and the time that your transaction was added to the block of a successful miner.  There are roughly 2000 transaction per block so for you to wait 30 minutes means that there were at least 6000 transactions in the queue before your transaction was added.