Jars, baseball is very simple (compared to cricket anyway).
Basics:
There are 9 innings. An inning consists of three outs from each team. An "out" is when the batter (aka batsman) is either touched by the ball while not touching a "base" (the bags on the ground) or the ball is caught without it touching the ground after being hit by the batter.
Points are scored by "runs". A run (worth one point) is scored when the batter runs around the diamond shape, touching each base (and not being touched by the ball) to return "home".
The ball is thrown by the "pitcher" (aka bowler) towards the batter, within a defined area. The batter's goal is to hit the ball far enough away to give him time to run to the base.
Each batter gets three "strike"s, which equals one out. A strike is when the batter swings at the ball and misses, or the ball is thrown into the "strike zone", which is a defined area between the chest and the knees (don't worry about this, it's negligible, but wanted to explain how someone can get a strike without swinging at the ball).
When the batter hits the ball, he runs to the first base, while the opposing team tries to get the ball and throw it to the "first baseman", who tries to touch the batter with the ball before the batter touches the bag. If they get there at the same time, the batter wins (aka "tie goes to the runner"). The batter is now "on base" and can be referred to as "the runner".
Then, the next batter comes up, and tries to do the same thing. The runner on first base must move to second base (there can only be one runner on base at a time), and they must go in order (first, second, third base, then home).
The runner may move when the ball is in play. The exception is when the runner "steals a base", which is running to the next base when the pitcher has the ball. This is risky, because the runner must be fast enough to beat the throw from the pitcher to the baseman.
This is very general, and leaves out a lot of nuance, but I think that covers the bases. (har!)