Of course, there are many styles of play and different FPS games, but there are a few rules that will help keep you in one piece whether it is a realistic shooter like the CoD series (in Hardcore, of course) or something as crazy as Unreal Tourament 3.
The Running Man Rule
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As simple as it gets - do not run in a straight line. Do not run in a pattern either. Moving forward, which you will supposedly do in any decent FPS, because I haven't seen one where you are completely immobile, yet, one must be as chaotic as possible. Jump more! No lines. No zig-zags. They're predictable and experienced players will just shoot you in the face or the location where he predicts your face will be in a split second. The predictable runner is just one more frag for the sniper! Don't be that frag! Jump and sidestep often! Step forward, jump, step left, jump, step left again, step forward, double jump. Keep it so random that your assailant can't guess where you will be - that is one key of getting shot less.

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An Open Target Rule
If you are in the open, you are dead. Use cover! All your movement must be from one cover to another one! While this is not as important in Unreal Tournament, it is a life-saver in realistic shooters. Find cover. Duck. If you stay in one place too long though, that is only a bit better than being in the open. Plan your route to the next piece of cover. Get there using the Running Man rule above. Proceed to own. You'll be surprised how many people completely ignore cover and just charge through the open, where you, the smart person, who has read this article and are now sitting in cover, can take free shots at them.
The Shoot Them In the Head Rule

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Simple as that. Whatever the game is, if it's worth playing, headshots kill people faster. It is simple, really - instead of spewing precious ammo over their bodies, fire in short bursts, when the crosshair is where their heads are, or if they are running predictably, will be.
There Can Be Only Team Rule

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People often tend to think of themselves as being invincible. Mostly, this happens after finishing some game's single player mode and then trying to play multiplayer. They grab their favourite weapon and charge, alone, to get the objective! And then, they get turned into a dead body, nicely filled up with bullets, or depending on the availability of explosive munitions, a fine red mist. You should always cooperate with your team, snipe if they need support, assist the charge in person if they need that. Do not wander around alone, because you can never secure all flanks just by yourself.