Fishing is a sport that is used in many games. Sometimes it's used for food, sometimes it's a skill, sometimes you fish to gain levels or to sell the product, or a game can even be solely about fishing. In a game like Runescape, the fishing is all about catching, selling, and healing. You also level up the more you fish, which allows you to catch more kinds of fish. In a game like Fishing Master it's all about the fishing; in Zelda, it's all for food and health.
Fishing as a genre
Fishing games can be a genre all to themselves. Games like Fishing Master, SEGA Bass Fishing, Cabela's Monster Bass, and Rapala's Fishing Frenzy are all games of the fishing genre. While some of these games go for more of a simulation experience, others go for more of an arcade experience. Some games have even gone so far as to include rod and reel game controllers, like Rapala's Pro Bass Fishing.
Fishing as a mini-game
Mini-game based fishing is probably how most casual gamers are familiar with fishing in video games. Fishing is most prevalent in role playing games such as World of Warcraft or Age of Conan as a crafting skill that allows players to craft items for sale or use in-game. This skill typically combines with a cooking skill that allows players to create meals or food-items that are useful in regaining health.
The simplest form of fishing is just an action that is activated by the player. An additional action may be required to pull up a fish once it is hooked. A more common fishing method allows the player to move the line and aim to hook a specific fish. Once the fish is hooked the player must reel it in while also fighting against the fish, and using its motion against it. Some games add line strength to this equation, forcing the player to ease off if the line is at risk of snapping.
An alternative to the 3 dimensional style is a horizontal style, seen in Breath of Fire III and Suikoden II. The fish is represented on a horizontal plane, and the player must keep a bar on top of them as they move back and forth in order to reel the fish in.