This section doesn't cover one rule, but rather a number of rules that apply to a baserunner.
It's understood that a runner must advance from first to home by touching the bases in order. It does not matter if your you have baseball cleats in different styles, your cleats have to touch the bases in order. Where the confusion comes in for some people is when the player must go back to their original base. In this case the player must retouch all bases in reverse order. This situation occurs most often in the following example:
Runner on first base. Ball is hit into the outfield and the runner thinks it's going to drop. He runs past second base and the ball is then caught by the outfielder. He needs to get back to first base before the throw in order to not be forced out.
The correct way for him to do this is to first retouch second base before running back to first. If he runs back to first without retouching second base, the correct call would be for him to be out.
The only exception to the rule is on a dead ball. If the ball goes foul for example, the ball is dead and the player doesn't have to retouch second on his way back to first.
There is no rule that states that a tie goes to the runner. Well at least not in those exact words. The portion of the rule in section 6.05 that applies to this states, "A batter is out when after he hits a fair ball, he or first base is tagged before he touches first base." Since the rule states "before", it is easy to assume that a tie does in fact go to the runner. But, since this is a judgment call you won't win an argument by yelling "a tie goes to the runner".
Two ruuners are not allowed to occupy the same base. If two runners are touching the same base, the lead runner is entitled to the base. Most coaches will teach their defensive players to tag both runners when they are occupying the same base. When this happens the lead runner is safe and the other runner is called out. With all things being equal (meaning I wouldn't have a player run around one runner without tagging him to tag the other), I tell my players to tag the following runner first. The reason for this is: If they tag the lead runner first, and the other runner starts to go back to the other base before he has been tagged, no out has been made. We now have a rundown and no out. As you know, especially in youth baseball, anything can happen at this point. If the defensive player touches the following runner first, he is out as long as the lead runner is still on the base. If the lead runner takes off then at least we will have a rundown to get the lead runner rather than a rundown where the players have to keep an eye on the lead runner.
There is one common misconception about overrunning first base. Some people want to argue that a player has to turn to the right (foul territory) after crossing first base in order to safely return to the base without the chance of being tagged out. This is incorrect as there is nothing in the rules that state which direction the player must turn. The rule states that the runner cannot be tagged out after overrunning first base as long as he/she immediately returns to the base.
Of course there is an exception to this rule, in that the runner can make no attempt to go to second base or he can be tagged out. This comes down to the judgment of the umpire and not whether the player turned into foul territory or not after crossing first base.
The other exception to the rule has to do with immediately returning to the base. If a runner overruns first base and thinks that he is out or that there are 3 outs, and proceeds to walk towards the dugout or his defensive position, he can be called out on appeal after the base or the runner is tagged.
Although it's not stated in many of the rule books, the reason for overrunning first base is based on the fact that the momentum of the runner doesn't allow for him to stop on first base. So keep in mind that if a player runs to first base and then stops on the base, he is not then allowed to overrun the base and get back without the chance of being tagged out. Once he steps off the base in any direction in that scenerio, he is fair game.
Some weird things can happen on this play, one being the first baseman attempting to tag the runner as he returns to first (believing he made an attempt at second) and the runner attempting to not be tagged as he gets back to first base. Although it may make him appear to be guilty of of trying for second if he attempts to avoid the tag, he should be called safe even if tagged, if he didn't make an attempt toward second.
Finally, some leagues allow a batter-runner to overrun first base on a walk or hit by pitch while others do not. Even though it shouldn't come up, it's a good idea to know how your league rules regarding this situation. Little league for example allows this.