One of those fun English idioms that has way too many meanings. Here are some of them:
1. To strike out can be to start a new venture, or to start on a journey; At daybreak we struck out on our own, hoping to find a source of clean water.
2. To strike out something is to cross it out (
like this), or to delete it. In legal proceedings, this is used to refer to removing something from the court proceedings, such as written material or a statement. The court ordered the claim for damages struck out.1
3. To strike out at someone or something is to attack it. When speaking of physical violence, this implies hitting and punching someone, but it is also used metaphorically, i.e. Soy producers strike out at unfair stereotyping. This is often worded as 'strike back at'.
4. To strike out is also to fail, or to fall out of favor with someone. This almost certainty derives from the sport of baseball, in which striking out is the worst way for a batter to go out. George Bush strikes out again.
5. And finally, baseball! There are many ways for a batter to be out; being tagged out or forced out, bunting foul with two strikes, etc. All of these mean that the batter has at least hit the ball, even if only poorly. To strike out is to have three strikes, that is, to have swung and missed the ball three times. Confusingly, you can strike out without actually being out -- if the catcher does not catch the ball, the batter can run for first base even if he has three strikes. (The number of strikeouts is a statistic kept on both the pitcher and the batter, so the number of strikeouts matter even when an out is not effected).
In baseball it is often written as one word, strikeout (it works better in the plural that way). There is more information on the baseball strikeout in that node.