Here's some of the games that are frequently dealt.
Non-wild card games:
Texas Hold Em High Half Pot
This is standard Texas Hold Em except the betting limit is half of whatever is in the pot. About once or twice a session, betting can really heat up. Early round betting (capped at 50 cents) very rarely gets many people to fold, so you have to have a hand that holds up against draws. It's more common to be heads up only for the river, and whoever is still in there has been drawing, so it's easy to lose by then even if you had them beat along the way.
This is basically the four hole card high/low 8 or better game where you must play two cards from your hand in combination with the 5-card board. This is one of the only "cards speak" split pot games, that is, players don't declare but show their cards. A major difference from standard Omaha is that the best low is a 6432A, not a 5432A (in other words, straights and flushes must play high). Betting is usually regular, but sometimes quarter or half pot. If there is no low, it can be tricky determining high hand values even when the board looks threatening, as players tend to stay in high with just about anything. If on the other hand you have a low but it can be beaten, it probably will be. Fold.
Three-Card Texas Hold Em High/Low Quarter Pot
This variation on Texas sees each player dealt three hole cards. The best high splits with the best low. Betting can be up to a quarter of what's in the pot. It's difficult to know what the best high hand is from reading the board, since a player could have a flush with a two-suited board, or a well-hidden fullhouse (let's say hole cards are QQ7 and the board at the turn is Q547, for example). As a result, the best hole cards to get are three solid low cards such as A23, A34, A26, 234, etc. It's not a qualified low, that is, "8 or lower," so three low cards that work together without forming a straight will often take half the pot. Hand values high are more risky here, with two pair sometimes taking it and other times very strong, well-hidden hands being out there. If the betting heats up you just have to use your intuition as to whether it's a battle for the low or high. And two low cards with no help high, such as A29 with a board of 34K faces 10 bad cards at the turn and river (three aces, three 2's and 4 5's, although the A2345 straight could win high). A very tough game to read.
Old West (Traditional 5-card draw)
What can be said that hasn't already been said about this ancient poker game? Bluffing, overbetting the pot and check raising can be common. I generally stay out of the way unless I'm dealt at least JJ or better.
Brag/Pippi Longstocking/Pippi Stopsmoking/etc.
This is the three card poker game dealt in the film Lock, Stock and Two Smoking Barrels. Three hole cards are dealt to each player. To open, you must match the ante. Betting is pot limit from then on. The best hand is three 3's, followed by three of a kind from aces down, followed by a three card straight flush, a three card straight, a three card flush, high pair, high card. Note that a three card straight beats a three card flush. Catching a flush or better is rare. Most bets are bluffs or semibluffs with a pair or an ace. But not always, of course. Since it's pot limit, it sometimes gets some action, but rarely from more than three players.
That's basically it for non-wild card games. The most common wild card games come in the next installment.