Affecting win percentages in turn based games
There are many different options:


Hex uses the "swap rule", which states that after the start player makes his first move, the other player can decide to switch colors and play as if he were the start player and had made that move. This means that the start player must choose a move that's not obviously great (like going directly in the center).
Magic the Gathering has the player who takes the first turn skip his initial draw, so he starts with one less card. Certain types of decks prefer to still take the first turn, while others prefer to draw the extra card. Similarly, Innovation has the first player only take a half-turn on the first turn. Or the first 2 players in a 4 player game.
Dominion has the loser of a game go first in the next game. This only works when playing multiple games in a row with the same people; but it's designed to be played that way. Dominion also has the second player win tie games if the first player got an extra turn (if he was also the player who ended the game). 
Many games give extra starting resources to players in later turn orders. Temporum, Castles of Burgundy, Agricola, and many others do this. Sometimes it is as simple as the start player receiving fewer than the rest, other times it is each player getting 1 more of something than the previous player. (So players will get 2, 3, 4, and 5 workers respectively in Castles of Burgundy).


Some games give later players a victory point bonus.  Go and Castles of Mad King Ludwig do this.  This can also overlap with the starting resources handicap in games where resources are directly converted to victory points at game end.

Many games only end when all players have had the same number of turns. Whether due to having a specific number of rounds, or having the game continue until the last player has a turn after the game-end has been triggered. Even in games like this, there may be a first player advantage, though there can also be a last-player advantages instead.
Catan evens out the setup by reversing the placement order in the second round of setup. In theory, this gives the initial player a great location and a poor location, while the last player gets two average locations. In this arrangement,
the second last player might be in the worse position, because the last player has the advantage of placing twice in a row.


Some games don't try and balance first player advantage at all and instead rely on player politics to do so.  These tend to be high luck games such as Munchkin or Fluxx.