GURPS, from Steve Jackson Games, is a generic RPG, which means it's not specifically made for superhero action. However, because it's generic, GURPS is designed to support virtually any character you can imagine.  This means that it's powerful and incredibly customizable character creation system is perfect for re-creating virtually any superhero, whether they're an established one from the comic books, are a completely unique one of your own invention.

​Because GURPS's rules tend to be more on the detailed side, this won't be the RPG to pick if you're looking for a very quick-playing, super-powered brawling system.  However, if you're looking for the kind of RPG where (for instance) shooting at a flying superhero/villain requires checking both the flyer's speed and the distance away, GURPS will offer exactly the kind of "realistic" combat you're looking for.

And because GURPS rules support virtually any character you can imagine, doing virtually anything you can imagine, GURPS is perfect for superhero campaigns that aren't just combat-focused: it's skill-based approach can handle a much wider range of non-combat interactions than many other superhero-focused games (which tend to simplify things by using fewer skills).

Character creation in GURPS is detailed, but slow.  There are no levels or classes, just points, which can be used to buy attributes, advantages, disadvantages, or skills.  Different campaigns start with different amounts of points, so that (for instance) a superhero can start out as a more powerful character.

Attributes

GURPS has four attributes: Strength, Dexterity, Intelligence and Health.  Strenght and Healthy just define the character physically, while Intelligence and Dexterity are combined with skills to make skill checks.

Skills

Skills are how characters accomplish things in GURPS; for instance to shoot someone you would use the "Guns" skill.  Skills are based on attributes, so for instance you might buy the Guns skill at a "Dex + 1" level.  If you later improved your Dexterity, you'd similar improve your ability to shoot ... or do anything else physical.

Advantages

Advantages cover any non-skill benefit a character might have, such as a high pain threshold, wizard training, or the ability to move objects with their mind.  Advantages can also be further be customized with enhancements or limitations, allowing you to gain more or less powerful versions, that cost more or less points as a result.  This can allow for a nearly any power you can imagine to be described as a GURPS advantage.

Disadvantages

Similarly players can also take disadvantages for their character, such as Code of Honor or Blind, to gain extra points.  All together this system allows for incredibly individualized character, including almost any you'd want to re-create from fiction or real-life (in fact, there are multiple GURPS books with stats for historical NPCs such as Ghengis Khan) ... but it can be a bit overwhelming to newcomers.

To succeed at an action in GURPS, players roll 3d6 and attempt to rull under a target number, which varies depending on the action.  Most of the time the actio will be a "skill check", meaning the player will have to rul equal to or less than the character's relevant skill. 

Critical Successes and Failures

If you fail a roll by more than 10 you critically fail the roll, and the GM decides what terrible fate results.  Similarly if you beat the roll by more than 10 you instead get a critical success, dealing extra damage or otherwise succeeding with flair.

A Note About 3d6 (vs. d20 or Other Dice)

3d6 results in a far more predictable distribution of rolls than a single die roll (typical in other systems).  Think about when you roll stats for a Dungeons and Dragons character: you usually get a lot of more 10's and 11s than 3's and 18's.  In contrast, every d20 roll has an equal (5%) chance of rolling a 1 or 20 as it does a 10 or 11.

This allows GURPS to have critical successes and failures, but have them be rarer and more dramatic, while leaving most rolls with fairly predictable/average outcomes.