Basically, I suck at aiming. I can really only do it semi-well with a mouse. Since the things they have you aim at are usually shooting back at you, shooters can be pretty stressful for me.
I just finished playing the excellent Mass Effect 2. That game sort of plays like a shooter, except for the vital ability to pause the game mid-fight to assess the situation and set up your powers to fire off when you unpause. The fights are still action-y, but they were much more laid back because you can always stop and figure out what all is going on. Fallout 3 had a really similar system, except that after you queued up your shots, you got to watch your character make heads explode in sweet slow motion.
I just got done replaying the first hour or so of Bioshock. This one's also a shooter, but no pausing and heavy on the atmosphere. Way, way more stressful. I don't think I've really come close to dying yet, and even if you do die, there's handy resurrection stations set up all over, so you just pop up again like a Buddhist on speed (with guns). So death, not so much something to worry about, yet not being able to stop and look around during fights makes all the difference.
I could play Mass Effect for hours at a time and end up pretty relaxed. Bioshock, I find myself feeling pretty tense after I'm done. I think I need to figure out that getting killed ain't so bad. Also need more ways to make heads explode.