'Tuning' your bow is the process of adjusting your bow so that arrows fly in a straight and consistent way.
For beginners, the best thing is probably to ask a coach or other experienced archer to help you set up your bow. Fine-tuning the bow isn't important in the early stages, because it's your technique that has the biggest effect on the quality of your archery.
As your archery improves and becomes consistent, learning to tune your bow will become a useful skill. A well-tuned bow will help you to achieve tighter arrow groups at different distances.
For indoor shooting, bow tuning isn't a big deal. As long as your bow is set up 'about right', you'll be able to achieve good scores. Basic setup for a new bow takes about half an hour.
Outside, it's a different matter. Your arrows might hit the target perfectly at 30 yards, but when you move up to 40 or 50 yards, they might start landing too far to the left or right. This can lead to a lot of frustration where you have to keep adjusting your sight both vertically and horizontally. But assuming that your technique is OK, this is a problem that tuning can fix. A well-tuned bow will shoot arrows that land centrally on the target at different distances, and you'll only need to adjust the sight up and down, using 'sight marks' that you've recorded. Tuning can also improve arrow grouping and reduce noise. A bow tuning session can take a couple of hours..
There are a number of guides to bow tuning available online, but the two main ones are from Easton and World Archery. Just follow whichever you find easiest to understand:
Easton Arrow Tuning Guide (download the PDF from the linked page)
Recurve Bow Equipment Tuning (Intermediate Level), from the FITA Coach's Manual (direct link to PDF)