Position simply refers to where you are in relationship to the beginning and the end of the action on a round of betting. If you are first to act, you are in first position. If you are last to act, you are in last position. Other positions are classified as early, middle or late depending on where they are, where early is close to first and late is close to last.
In hold'em, for example, the holder of the button is in last position while the player to her right is in late position. The first person to act after the flop would be in first position, with the player to her left being in early position. In hold 'em, these positions do not change for the duration of the hand, except that as players fold, other players may end up assuming their former position; the first player to act on a betting round is always considered to be in first position. Likewise, when the person in last position folds, the next-to-last player now has the last position. In Hold 'em, getting last position in this way is called "getting the button"; if it was achieved by the player betting and inducing the players between her and the button to fold, it is also called "buying the button".
In a game that does not use a dealer button, like seven-card stud, the holder of the low card on 3rd street is in first position with the bring-in. Of course, on later streets, the first position changes, going to the player with the high hand among their upcards.
It's always best to be in late or last position. That player always has more complete information about the actions of her opponents and can make more informed decisions, and can often close the action if she chooses.