Back in the 'nam days, didn’t US fighter planes many times have the two-man ‘backseater’ setup? But now you only see the one man setup.
So what happened – wouldn’t it be better to have two heads instead of one, making the crew much more efficient?
In a cutting-edge, high-performance machine, there’s not much room or weight-carrying capacity to spare – everything is kept as lean and mean as possible.
You can keep improving the electronics, and the design, and to some extent the materials, faster than you can improve the human pilot/crewman.
In particular, modern computing power can enable one pilot to manage the workload that pilot and radar intercept officer used to split. And software weighs nothing.
An individual human comes with a lot of requirements to stay alive and functioning in the high altitude, high-acceleration environment of a modern fighter aircraft: ejection seat/parachute, pressure suit, bottled oxygen, communications, separate instrumentation, heating/cooling systems, and so on. All that gear in turn requires additional fuel (to reach the same range); equipment, human, and fuel impose additional weight cost, both putting a strain on the airframe and reducing the thrust/weight ratio.
So, while there is probably some advantage to a second human, particularly in being able to pay full attention to two different things at once, the cost is high and what the second human brings to the party can be partly replaced by lightweight technology.
Since air superiority fighters are competing with each other in a ruthlessly Darwnian environment, the second human is being squeezed out.
Improvements to Unmanned Aerial Vehicles may soon squeeze out the first human.
Sailboat
This won’t be a definitive answer, but IMO second crew became more important when fighter tactics shifted from close-in dogfighting to missile attack at a distance. A second crewperson occupies space and costs weight and fuel that will reduce the effective payload of the aircraft, so if Vietnam-era fighters were more likely to have a second crew, this would be because the workload involved with target aquisition and attack demanded it.
In recent years, there seems to be less of a need for a back-seater due to much more sophisticated on-board systems that require less pilot input, as well as the use of AWACS aircraft that take on much of the target acquisition and monitoring role.
On the contrary. WWII warbirds were mostly single seat machines, with a few exceptions who had rear gunners. Twin seaters came in vogue when someone was needed to read the radar, assign missile targets etc… while the “real” pilot was doing pilot stuff, like making sure the ground stays below the plane rather than in front of it. The F-14 was basically designed as a moving platform for the longest-range missile ever devised, yet it’s a famous two-seater. Talk to me, Goose !
That’s my take on it as well - why fly a radar guy and his oxy supply, when there’s a 747 full of top notch radar guys already feeding your aircraft non-stop with data, telling you who to shoot at and when, and who direct your own missiles in flight ? No need for that in an air-to-air fighting plane anymore. As to watching multiple bogeys and reporting relative situations, that’s what wingmen are for.
Same goes for air-to-ground combat : there needed to be a WSO to read the maps, compare it to ground radar data, read the IR/TV images to ID the target, keep the laser beam on target etc… but today, with JDAM technology, even iron bombs home in on a preset GPS target, so that the pilot merely has to get to a nav point at pretty much any altitude and speed, and the bombs will still hit within a few meters of their mark.
Pure ground pounders like F-15 Strike Eagles, Tornadoes, the israeli F-16I variant etc… still have two seats, and are used on unknown location targets (like constantly moving SAM sites, ships, tank columns etc…) but for general purpose bombing the crap out of buildings, a single pilot is more than enough.
As stated above, the basic answer is automation removed the need for the backseater (weight really didn’t have much to do with it-- saving weight is a nice bonus in fighter-bomber design, but saving the extra cost to train and employ another pilot as a backseater, THAT was a real coup).
Still, as Kobal2 notes, there are plenty of two-seat aircraft still built. Usually as trainers, but many of those trainers have dual-missions to serve on the frontlines when/where necessary.
Also, it’s not just ground attack that likes to use two people: the U.S. EA-18G Growler (the electronic warfare version of the two-seat F/A-18F Hornet) requires two crew, one pilot and one crewman to work the electronics.
On the contrary? Isn’t that exactly what El_Kabong said?
Automation, automation, automation.
Note that strategic bombers still generally require multiple crewmen - the B-1, B-2, B-52, etc. The move back to one-man fighters is also a recent one; the F-14 and F-15 were two-man fighters; their replacements, the F-22 and F-35, are also the replacements for their cheaper single-seat counterparts, the F-18 and F-16.
The second pilot kept talking about his girl back home and all his plans for after the military, so naturally he was eliminated.
It’s not just fighter planes that have seen crew reductions due to modern avionics. Consider the B-29 bomber used during WWII, which had an 11 man crew. Compare that to the B-2 stealth bomber, which has a 2-man crew. It goes twice as far, twice as fast, and can carry more than twice as many pounds of ordinance.
(I suspect they could fly with a single pilot if they flew shorter missions. Most of their missions in Afghanistan, and Iraq take off from and return to Whitman AFB in Missouri, a round trip of over 30 hours.)
I used to hate the way that guy would tape a picture of his sweetheart onto the controls. That really grinds my gears.
Note that there’s a widespread belief that the trend will continue, with 1-man fighters giving way to unmanned versions which are predicted to be cheaper and much more agile.
More like a dream, I doubt that the powers that be will allow an autonomous drone with live weapons. The closest your going to see is a variation on the predator with a human on the other end deciding when to engage a target.
Mind you the tomohawk cruise missile is exactly what I just described as what wont happen , but that is preprogramed before the strike and is a one use weapon.
Declan
Xema said nothing about autonomous. They already have Predators/Warriors armed with 2/4 (respectively) Hellfire missiles, and the Reaper is a new multi-munition platform that can carry 14 Hellfires.
Thats me jumping to conclusions , but the wording was sorta ambiguous when xema mentioned
So autonomous kinda jumped out at me but was never strictly stated.
Apologies all round
Declan
I guess you never saw this documentary on why no-one will allow an autonomous fighter?
Actually I own that movie , at least my copy of it. It was a decent portrayal of a hollywood fighterplane 
Declan
Alas poor Goose.
Not to worry. The guy with a young wife and kids, or a sweetie he’s going to marry, always gets it pretty early on, right before the guy who’s flown his quota and is about to be transferred home but wants to fly just one more mission.
A belief so widespread it’s addressed in post #2. 
Well, as R2 and Dak can attest, riding bitch for Luke is a sure way to get shot in the head.