Everything I've covered so far on this page has been about video games, but that doesn't mean that they are all there is to gaming. Gaming covers a vast spectrum of possibilities for competition and cooperation. A Gamer can play video games, but they may also play board games, role-playing games, miniatures, card, and tabletop games.
Everyone at some point has to have played a board game. Candyland, Monopoly, RISK, checkers or chess, any of them count. Having played it once doesn't make you a Gamer, nor the occasional game, but it's a great introduction to the Gaming mindset. Most people don't get Gamer-fanatic about board games, but there is the possibility of it leading to becoming a Gamer.
When board gamers get to the Gamer level they tend to be into the more complex and obscure games, many of which can be a lot of fun for non-Gamers as well. If Monopoly has lost its luster, or you want to step it up from RISK, you could try out Settlers of Catan or Axis and Allies.
Settlers of Catan, or "Catan" as it is sometimes called, is a game based around an island. Each player builds settlements and roads to gain resources and "Victory Points" with which to win the game. It is very easy to pick up and every game is different, due to different board setups and die rolls. It's also a very social game because the game hinges on making trades with other players to gain resources that you may not have.
Axis and Allies is a World War II based game which has many similarities to RISK, but many more complexities as well. You have to keep up an economy and spend your resources wisely on different unit types. It can be a long game, but then again, RISK isn't exactly short either.
Catan belongs to a sub-group of board games often referred to as German-style board games, because many of them come from there. They are often more about social interaction or building things than just moving pieces around a board, as well as being about scoring rather than eliminating other players. Other games of this type are Carcassonne and Ticket to Ride. These games are relatively easy to pick up and have won many international awards. These are wonderful games for families or groups of hobby-gamers. They are simplistic, yet deep.
Role-playing Games, or RPGs, are games in which the player takes on a character and develops them over the course of an adventure. The best known of these is probably Dungeons & Dragons, although many more exist. There's the World of Darkness series, which covers vampires and werewolves, Call of Cthulhu, based on the mythos of H.P. Lovecraft, and even a Star Wars based RPG.
These generally require a group to play, with one person acting as "Dungeon Master." The DM controls the world around the players and moves the story forward. The players generally play out of books and use paper and pencil to keep track of their progress. Battling is generally done with dice.
From what I understand and my own experience, it can be hard to just jump into playing these sort of games with little or no experience, but it is also not particularly difficult to find people who have played and might share an interest in teaching new people to play. Wizards of the Coast (D&D's publisher) also has many tools to help people get started, as do the makers of most of the other games.
The above games revolve around a core set of items. Board games come in one box, and RPGs books and dice, but the next two types of games require constant additions to remain competitive. Just a warning, this can get expensive (but totally worth it, should this be your type of game).
Miniatures gaming is a form of tabletop gaming where you take an army of plastic or pewter figurines and battle them against another person's army. The grandfather of these games is Warhammer, which is based in a fantasy world of orcs and elves. Warhammer 40K moves that world into the space-faring future with the same ideas. You take a race of creatures, build an army of minis, and battle them across a table using a series of rulebooks as a guide. (If you live in the Amarillo area, early afternoon on Saturday is a good time to check the local shops if you're interested in games like Warhammer.)
Customizable Card Games, or CCG's, are not like Uno or Poker. They use special decks of cards that are made by the player out of an available abundance. Games like Magic: The Gathering, Pokemon cards, and Yu-Gi-Oh have thousands of possible cards and the player builds a playable deck from what they have. In order to get better you have to not only come up with new strategies, but also acquire new cards. This can quickly get expensive. There are, however, many tournaments and groups at which you can win and trade cards.