David Landry wrote this in 2007. I have refreshed it.

The out is the cardinal sin of baseball. Each team gets only 27 outs – three outs for each of the nine innings. It is the goal of every baseball player to minimize the number of outs he makes.

Batting
Batters have the most chances to make outs. This is why the most frequently used statistic to judge batter success is the on base percentage (OBP) – it measures the percentage of time the batter reaches base without making an out. Here are the ways batters make outs:

The batter hits a ball and it is caught by a fielder in fair territory before it touches the ground. Sometimes this is called a line out, when the ball is hit straight or a fly out when the ball is hit on a high arc.
The batter is tagged before reaching base safely.
The fielder touches first base while holding the ball before the batter can touch the base.
The thirds strike is caught by the catcher. If the catcher fails to catch the third strike, he must tag the batter with the ball or throw it to first before the batter reaches base.
An infield fly when there is a runner on first base and there are fewer than two outs. This is called the infield fly rule. The purpose is to prevent the fielder from purposely dropping the ball, allowing him to recover it quickly and make a double play.
A bunt that lands foul on the third strike.

Running
The runner can make outs in several situations. As long as the runner is touching the base, he cannot be tagged out, however, he needs to take risks by taking a lead. This may be a prelude to an attempt at stealing a base or it may be an attempt to reduce the number of steps required to make it to the next base when the ball is hit. Here are the ways base runners can make outs:

The runner is tagged while standing off the base.
The runner runs outside the baseline.
The runner interferes with the fielder (my favorite example of this was when A-Rod batted the ball out of the glove of Bronson Arroyo, who later referred to it as a “little league” move).
The runner is hit by a batted ball.
The runner fails to touch the bag (tag up) after a ball is caught, when running to the next base. In this case, the fielders need to tag the base the runner was on to make an out.
The runner has another runner behind him and is forced out when the base he is running to is tagged.
Two runners are on the same base.

The Double Play
There are situations where two outs can be recorded in a single play. The most prevalent example of this is the GDP (Ground into Double Play), where there is a runner on first and the batter hits a ground ball. The fielders tag second base, then throw the ball to first for the second force out.

The Sacrifice
This is arguably the exception to the rule about making outs. A sacrifice situation is one where a player makes an out to cause a specific strategic outcome. The sacrifice fly is the simplest. With a sacrifice fly, there is a runner on third base and there are fewer than two outs. The batter hits a deep fly ball, allowing the runner on third to “tag up” (the runner must wait until the ball is caught before leaving third base) and score before the outfielder can throw the ball to home.

The other key sacrifice is the sacrifice bunt. In this case, the object is generally to move a runner into scoring position (second or third base). This is generally done when there are no outs and a runner is on first. In this case, the batter bunts (allowing the ball to hit his outstretched bat), causing the ball to dribble along the foul line. This causes the first or third baseman to have to run toward the plate to field the ball.