If I am playing a four man free for all, when do my artifacts untap using Unwinding Clock's ability? I'm having trouble understanding. The card says "each other player". If I am player one, are players two three and four "other players" or is player three the other player since it says "each other player"?
You untap your artifacts during player two's untap step, player three's untap step, and player four's untap step.
This isn't anything that's in the comprehensive rules or anything, it's just the meaning of the words on the card. "Each other player" could be explained as "each player who is not you" or to be really clear "every single player who is not you". None of those players is you, so you untap your artifacts during each of their untap steps.
I think you might have been wondering whether "each other" meant alternating (yes 1, no 2, yes 3, no 4...) like "every other" sometimes does. But "each other" doesn't ever mean that, so there's no ambiguity here!
You can see this demonstrated indirectly a few places in the rules, for example:
103.4. ... First, the starting player declares whether or not he or she will take a mulligan. Then each other player in turn order does the same. ...
Obviously that applies to all players, not alternating players! And of course, all the cards that say "each other player" wouldn't really make much sense if they meant alternating starting with you - in a two player game, that'd be just you.
Each other player always means every player that is not yourself, not to be confused with alternating players. I've not come across a card that says this but I think this is what is confusing you?
Cards usually use each other player to signal every opponent http://gatherer.wizards.com/Pages/Card/Details.aspx?name=grave+pact, and target opponent to signal just one opponent. http://gatherer.wizards.com/Pages/Card/Details.aspx?multiverseid=227064