From Unreal Tournament '99
This Rocket Launcher isn't the best, it's projectiles don't do nearly as much damage as they could but there's the multiload function, two different firing modes, and two different firing spreads. Those make this one of my favorite guns of all time.
Teleport gibbing was one of my #1 ways to get kills in UT and it was very satisfying.
One of the first twitch reaction auto release melee weapons I really enjoyed using. You could also jump with it.
The dual firing modes and combine area of effect explosion attack are thematic and terrifying. Also hours and hours of instagib.
From Unreal Tournament 2k3
Graphically pleasing to use and annoying to be shot with, this gun slowed your targets down and even on semi close hits would cause damage.
You haven't played Quake til you've played railgun instagib for a few hundred hours.
From Tribes: Ascend
100% inheritance of your movement speed made this gun make so much more sense to fire for me and I immediately fell in love with it.
This and the Zeus are so satisfying to get kills with.
While they're entirely too powerful in T:A I love the way that these things fire and the readable delay between their launch and hit.
The Jakal used to be my only choice for primary as Infiltrator before the explosives it launches were given a very very small radius for effective hit. It launches 3 sticky explosives quickly in succession if you so choose, cannot be reloaded until the current explosives are detonated and it is very satisfying to stick enemies with.
Once you've got it to where it does more damage to armored targets this thing is hands down my favorite weapon from Tribes: Ascend. Long distance bombardment shots with this mortar are practically an art form. The 3 second prime is wonderful and the arc it leaves behind the projectile is just pretty.
From Goldeneye N64
The RCP-90 is a fire rate bastard with a massive clip, its bullets could penetrate doors and some walls.
I've always loved remote mines because of Goldeneyes', the simple control of throw and explode midair was very satisfying and quickly forced my friends to disallow them as a weapon choice for any matches ever. (Then I had to learn to shoot proximity mines in midair.)
From Perfect Dark N64
This thing was one of the strangest sniper rifles I have ever used. An alien gun it sees through walls and fires perfectly straight. Essentially a sniper rifle that you would use from a closet or even a cubicle to shoot someone within the guns range in any direction no mattter how many walls were between you and them. A classic gun from a classic game.
From Turok N64
The unmistakable dentist drill-esque noise bobbing and weaving through the air followed quickly by bone and flesh being drilled through made this gun a satisfying launch it and forget it gun. It homed in on targets as best it could but was a semi-dumb homing projectile that would bounce in to walls a few times and simply drop to the ground out of fuel. Was an instant kill if it bored in to a skull though. Terrifying.
I love pretty much any bow with realistic drop and speed of arrow- I really like the bow from Farcry 3 & Blood Dragon and Rust as well.
From Dirty Bomb
The slash main fire and lunge alt fire of this sword are excellent and what I now consider the standard for any melee weapon in an FPS.
The remote mines work very similar to those in Goldeneye, smaller blast radius and have a hard limit of 3 placeable at one time and an 4 second recharge on each stock to a cap of 2.
This thing is a beast, it's not unrealistic like TF2's demoman's launcher. The grenades it launches all prime at launch and have a 2.5 second fuse and touch detonation. The projectiles are pretty small so hitting someone directly is pretty hard. I love the way that they bounce and spin.
I don't know how you couldn't love this thing the first time you use it. It's terrifying and powerful, while not always an instant kill it's a very potent psychological weapon as well.
I really like the way that it semi-homes in on targets, always loved the look of it too and the psheew~ noise the needles make. I thought that it should do bonus melee damage based on how many needles were loaded in it.
As far as items are concerned:
Damage Amplifier from UT99 - It's borderline unfair, but feels so good.
Mega Health and Quad Damage from Quake - Both announce globally to the server that someone has picked them up Mega Health gives you 200 health that ticks down 2-3 hp every second till you're back to 100 and Quad Damage makes every single gun terrifying, even the crappy chaingun.
Overcharge from Destiny and Team Fortress 2 - Both for different reasons. The Mediguns "Ubercharge" gives invincibility to the target and bonus damage, a really cool cooperative skill. Overcharge in Destiny unlocks your super power and allows for you to dish out some pure damage or even revive yourself.
Overshield from Halo 2 (the 300% bonus) - This also looks pretty fucking awesome.
The Stealth Pack (so long as it's used with throwing knives) in Tribes: Ascend - (This removes footstep sounds. I love the full invisibility and psychological damage that the stealth pack in T:A causes)