The question what is the best fighter in the title of this thread is too hard to answer. The Cold War spanned 50 years and there was a rapid pace of development throughout, so one type or another dominated for periods of time but it is impossible to determine which is the "best" as it means comparing types inter-temporally. It would be more useful to break the period down into 10 year or generational chunks and nominating a fighter from each. Somebody else can do that if they want.

 

The different question raised in the body of the first post, which is the most influential fighter is easier to boil down to two types, the Mig 17 and Mig 21 as operated in Vietnam (the Mig 19 could also be considered to have played a more minor role). Their success against a better armed and trained foe proved to the world that the path that was being taken at that time of fighters that emphasised pure performance and all missile armament was wrong. This led to the revolutionary teen series designs, particularily the F-16, which emphasised features necessary to win the WVR fight such as excellent manouverability, gun armament and frameless cockpit hoods. The success of the Russian designs also led to the energy manouvering revolution in fighter pilot training.

 

This legacy was dominant for almost two decades after the end of American involvement in Vietnam, when during GW1 modern command and control methods, and more reliable missiles were proven adequate enough for fighters to be able to rely on their BVR weaponary to a greater extent. The Mig 17 and 21's legacy does however still have an important influence on modern fighter design, training and operations.  

such a wide range brings so many aircraft, or iterations of a aircraft into scope...

The title says "best fighter" vs your later "most influential".  To cite a few influential fighters of the period:

 

1950s Mig15 as the aircraft that most influenced everyone's design at that time as well as later developments of that line till the 1960s Mig21.  In the 1960s one could certainly mention the MirageIII, F-4, and Mig21 but I'd personally mention the Mig25 as most influential as it was the aircraft that scared the west and most influenced future design especially the F-15.  For the 1970s through 1991 in my view it's no contest in the F-16 being the aircraft that even today one uses as a baseline for comparison in so many areas.

 

For overall most influential I'd pick the F-111.  This was the F-4 replacement for both the USAF and USN.  It's failure as a fighter led to the the F-15, F-14, Mig23, Tornado, and influenced other designs as well.  It ended up killing other aircraft like TSR-2 as well.  

I think it's too broad.  AGA is right--we're talking a time period from 1945 to 1991.  Aerospace technology just leapt all over the place during this time.  Figure it this way: when the Cold War began, both sides were using piston-engined fighters armed with machine guns and cannon, and limited to either ground-based radar intercept or more often, their eyesight.  When it ended, aircraft were jets going Mach 2, had missiles, used their own radars, and could kill opponents beyond the limits of human vision.  It's apples to oranges--basically, RedParadize's "USAAF vs. USAF" scenario.

 

If you divde it up, though...

 

Korean War: the F-86.  The MiG-15 comes very close, but the F-86 was the better aircraft--admittedly due more to its better pilots.

 

The early 1950s: the Hunter.  I admit to some personal bias here, because the Hunter was a good looking fighter and held its own with opponents.

 

The late 1950s: the F-8 Crusader.  When you're out of F-8s, you're out of fighters.

 

Vietnam War: the F-4 Phantom.  As much as I like the F-8 and the F-105, the F-4 was by far the most versatile.  Without ROE, the VPAF's MiGs would've never gotten very far off the ground.

 

Arab-Israeli Wars: you'd have to give it to the Mirage III...though as I mentioned on the Mirage III thread, that may have been a case of pilots overcoming the Mirage's shortcomings than any particular advantage of the aircraft itself.  (The same could be said for the early F-4s, though.)

 

The 1970s: still the F-4.  I can't think of any other fighter during this time period that comes close. 

 

The 1980s: the F-15.  If FS is going to claim that the F-15 never fought against a "symmetrical" opponent, then I'll make the same claim for the Mirage III.  Maybe the Israeli F-15s were up against Arab hamburgers, but the fact remains that the F-15's flight performance utterly terrified the Soviet Union.  I've read declassified articles in Red Star and Soviet air force papers from the 80s, and it generally consists of "Hurry up and get the damn MiG-29 and Su-27 ready, so this thing doesn't murder us over Central Europe.  Oh, and we might want to start training our pilots to show some initiative, too!"

 

If I had to choose one from the whole subdivisions above, it would have to be the F-4.

Right on target, Senty.  From 1980 onward, about the best the Soviets could have hoped for in the air over Germany would have been to tell Frontal Aviation to sit it out, declare the whole sky a free-fire zone, and hope their strategic and tactical air defense networks last long enough so the USSR didn't have to resort to nukes to stop NATO.  A big part of the reason for that was our Eagle drivers driving Eagles.

 

problem with analysis of Soviet Tech at the time was that the info wasn't as readily available. The west would have to wait until a disgruntled Soviet pilot would sfly his plane to Tokyo, or the Israelis would share one of their captured aircraft.

 

We didn't know how bad the MiG 23/27 was until well after it was in service. This influenced a lot of the western development to counter the percieved threats. And subsequently western design/manufacturing raised the bar well above Soviet development.