Fishing is a skill in The Sims 3, one which is greatly aided by the "Angler" trait.  There are several dozen kinds of fish which can be caught, including those which were introduced with the World Adventures expansion pack.  

Each fish has its own bait that can be used to catch it, and learning those baits is a skill in and of itself!  You can learn the baits by reading the fishing books.  Much like the recipe books, the fishing books are surprisingly expensive single-use (i.e. consumable) books which disappear after a Sim reads them.  Your Sim can also learn the bait by using something for bait, and catching something with it.  

There are two basic habitats for fish: salt water and fresh water.  Just like in real life, you find different fish in each.  (Except for the Alley Catfish, which can be found in either habitat.)  Additionally there are a few specialized locations for specific fish.  The Robot Fish, for example, can only be caught in the pond behind the Science Facility.

Each fish has a skill level that your Sim needs in order to catch it.  Fortunately, it's pretty easy to build the fishing skill - if you tell a Sim to fish, they will fish until they are on the verge of collapsing from exhaustion.  Unlike other skills (painting, e.g.) your Sim is never "done" fishing.  Although the game may nudge you with a pop-up message that says "The fish don't seem to be biting here" or notify you when you have run out of bait.

What can you do with the fish you caught?

Cooking
Drag them from your Sim's inventory into the fridge, and they will be used in any recipe that calls for them.  Just like with produce, fish never go bad - either in your inventory or in the fridge.

Selling
You can sell all your fish at once by going to the grocery store.  Or drag them over into the sell box on your Inventory tab to sell them one by one.  The better the fish, the more money it's worth!

Fertilizer
Fish are an excellent fertilizer for gardening.  Some fish make better fertilizer than others.  Even within that category, the better the fish, the better it works as fertilizer.

Mounting
Click on the fish in your Inventory tab and send it away for mounting.  These can make a nice display, although most Sims catch so many fish so quickly that it's impractical to mount them all.

Aquariums
You can buy a fish bowl, and put one fish into each bowl.  It's a giant pain to keep them all properly fed and not dying.  But some Sims have that Lifetime Wish of "13 perfect aquarium fish," so there you go.

Stocking Ponds
Once you have 10 fish of any given type, you can stock a pond on your property.  If you drag 10 fish into that pond, the pond will forever be stocked with an infinite number of them.  (Pretty handy for, say, Death Fish!  And if you stock a pond with Death Fish, you can fish them out of that pond at any time - no more having to wait until 11PM!)