First, make a fishing pole, by putting 3 sticks and 2 lengths of string in the crafting grid (you can get string from spider webs or tripwires).
Once you've cast your fishing rod into water, keep an eye out for a swirl of the water around your bait. This should happen every 15 seconds or so, and shows that something has taken your bait (usually a fish, but can even be boots sometimes which is handy!) so start reeling in your line straight away.
If you catch a fish, take the time to cook it first because you satisfy 3 hunger points with cooked fish (uncooked is only 1). There's standard fish you'll catch quite a lot, and if you've got the latest version of Minecraft you can even score some salmon, clown fish or puffer fish!
Sometimes you'll catch rubbish like rotten fish. On the flip side, you can even pick up some treasure!
Tip: every time you cast your fishing line, your fishing pole will degrade a little, moreso if you hit an object like the ground (in shallow water) or an animal. Stick to fishing in deep water and it will last the longest, or enchant the fishing pole to halt the degradation.
Warning: don't eat puffer fish raw. It will make your character sick for ages!