Normally the shell waits for a command to finish. Put & at the end and it runs in the background instead, so you get the prompt straight back while it works. The shell prints a job number in brackets and the PID.
$ tar czf big.tar.gz photos/ &
[1] 5120
$ jobs
[1]+ Running tar czf big.tar.gz photos/ &
If you start something in the foreground and wish you hadn't, press Ctrl+Z. That pauses (suspends) it and gives you the prompt. Then bg lets it carry on in the background, or fg brings it back to the foreground. With several jobs, use the number: fg %2.
jobs lists this shell's jobs. They belong to the terminal, so closing the terminal normally stops them. To keep a job running after you log out, start it with nohup (no hang-up): nohup ./long-task.sh &. Its output goes to a file called nohup.out.
For anything you'll come back to over SSH, tmux or screen are worth learning: they keep a whole terminal session alive on the server, and you can reattach to it later from anywhere.