There's the diamond on which the teams play, there's the offense (batters) and defense (pitcher, catcher, infielders, outfielders), and such. Let's talk about base runners.
Once a batter gets on base (hit, error, walk, hit by pitch, dropped third strike, catcher's interference, or as a pinch runner) he can pretty much do anything he wants.
His job is to touch all the bases and make it home, scoring a run for his team. And, the team with the most runs wins, so that's a good thing.
Now, only one runner can be on base at the same time. Legally, that is.
Two runners may not occupy a base, but if, while the ball is alive, two runners are touching a base, the following runner shall be out when tagged. The preceding runner is entitled to the base.
In fact, there was once almost a time when three runners were on the same base at the same time. But that was illegal, and caused all kinds of confusion.
In 1926, the Braves (back then, the Boston Braves) were playing against the Dodgers (back then, the Brooklyn Dodgers). And, with the score tied, strange things happened:
The Braves are at Ebbets Field with Brooklyn's Hank DeBerry on 3B, Dazzy Vance on 2B, and Chick Fewster on 1B, when Babe Herman drives a vicious liner against the RF wall that caroms back towards the second baseman, Doc Gautreau. DeBerry scores, but Vance initially holds up, then rounds 3B headed for home. Fewster stops at 3B. Gautreau throws home and traps Vance, who heads back to 3B. Herman slides into 3B joining the crowd, as Fewster steps off. Herman is ruled out for passing a base runner. Fewster, thinking he's out, too, walks off with Babe, and gets tagged out. Vance, still on 3B, later admits it was his fault. Poor Herman, who thought he had a triple but merely doubles into a double play, gets the blame.
After a foul ball, the base runner must retouch his base, but otherwise, he isn't required to stay on base. He can do most anything he wants. If he decides to run the the next base, he can. But, if he's tagged while off a base, he's out.
Running to the next base, without benefit of a hit baseball, a wild pitch (the ball gets passed the catcher, and it's the pitcher's fault) or passed ball (the ball gets passed the catcher, and it's his fault), or an awarding of a base (walk, hit by pitch, balk), is called a "stolen base." And any base ... except first ... can be stolen.
Actually, according to baseball legend, a player once ... and maybe twice ... stole first base. But it wasn't a good idea then. And it's really not a good idea now, since it's illegal.
Any runner is out when --
(i) After he has acquired legal possession of a base, he runs the bases in reverse order for the purpose of confusing the defense or making a travesty of the game. The umpire shall immediately call "Time" and declare the runner out;
According one version of the story, an early American League game between Cleveland and Detroit had Germany Schaefer stealing first base:
Playing Cleveland, the Tigers had the speed Davy Jones on third base and Schaefer on first in the ninth inning and tried a delayed double steal, but the catcher, Nig Clarke, didn't throw. Schaefer ran back to first base on the next pitch, then shouted across to Jones that he was going to second base again. The catcher, rattled, threw down to second and Jones scored the winning run. That tactic was then outlawed.
Oh, by the way, remember the three-on-third story? You may have picked up that one of the base runners was out for passing another runner. You can't do that.
Any runner is out when --
(h) He passes a preceding runner before such runner is out;
In fact, one of the most famous games in history included a situation of a runner passing another runner. And of of those runners was one of the most famous ball players in history.
On May 26, 1959, Pittsburgh lefthander Harvey Haddix was pitching a perfect game. Not just 9 innings of perfect ball. Not just 10 innings. Or 11. Or 12. But into the thirteenth inning, Haddix had not allowed a single Milwaukee Braves player to reach base.
Braves pitcher Lew Burdette had allowed 12 hits, but no runs. So the game continued into the 13th.
A Braves runner finally reached base on an error, blowing the perfect game, but keeping the no-hitter alive. With one out, Hank Aaron was walked, and Braves first baseman Joe Adcock hit a home run to end the game.
Only, it turned out to not be a home run.
You see, Aaron didn't realize the ball had cleared the fence. He thought it was still in play, and the game over because the winning run had scored ahead of him. So, after he reached second base, he left the field.
He shouldn't have done that. Because Adcock passed Aaron's position on the base path when he reached third base, and, by rule, was out.
Since the last base he legally touched was second base, he was credited with a double.
So, it's not quite as simple as a base runner being able to do what he wants. But he pretty much can.