Air Hugging

Male characters show discomfort expressing emotions, particularly affection towards another man. Often results in a "hug" with no physical contact, an obviously awkward hug, etc. There's also the version where females can hug with their bodies close but guys hug while their lower halves are far away, so they have to lean over more to hug.

Examples of Air Hugging include:

Comic Books

Elf Quest uses the "lower bodies separate" version near the end of Fire and Flight, the very first graphic novel. This despite their Everyone Is Bi policy.
A particularly confusing example comes from Young Avengers, where Hulkling and Asgardian, who are a pair of boyfriends, are frequently shown hugging with 45-degree torsos.

Film

One of the older examples in movies, Grease.
An excellent more recent example is a 'hug' between Will Ferrell and Adam Scott in Step Brothers
In The Lord of the Rings, Legolas and Aragorn have a habit of placing one of their hands on each other's shoulder and looking at each other for a moment.
Parodied in Groundhog Day where Bill Murray turns his daily meeting with "Ned" into an extended and and quite uncomfortable hug.

Live-Action TV

Everybody Loves Raymond, "What's With Robert?"
Chandler and Joey did this on occasion in Friends.
- They also lampshaded and averted it in the finale when they decide on the "lame cool guy hand shake" over the "awkward hug", but end up giving each other a real hug, anyway.
Adrian Monk and his brother Ambrose did this when they met for the first time in years in season 2. Although, both those guys also just have a problem with physical contact in general.
In the episode of Will and Grace guest starring Matt Damon, his character (who is pretending to be gay) does the second type to Jack. He gets out of the suspicion by claiming that by having his pelvis away from Jack's, he can then check out Jack's shoes.

Newspaper Comics

In a Dilbert book on etiquette, Dogbert discusses Air Hugging, saying that it is not the best time to reveal your sensitive side, and not to let people see you get the creeps afterwards.
Discussed in this^{[dead link]} Herb And Jamaal strip.
Zits had a strip (seen above) where Jeremy air hugged his dad on his birthday, with his mom commenting "Happy male birthday, honey.".

Real Life

There's always the patent "reaching out patting each other on the back we're-not-gay-honest hug" which seems to be endemic to all males of a certain age when formality dictates some form of physical embrace.
A related version to this comes from the infamous instance of the 'Christian Side Hug'. It's a rap. About how hugging frontwise leads to... y'know what? Just google it.
This has also been referred to as "A-Frame hugging."
There's also a thing where guys look like they're going to shake hands, but they instead each put one arm between their bodies and get close while pounding on the other guy's back with the other hand. Also known as a Man Hug
- Which has also a habit of showing up in situations where protocol demands you shake hands but emotional/personal background demans you hug.
The "Distance Hug" can also be used when a person is sick and doesn't want to pass germs.
- Also if one person has a really nasty sunburn and the others want to show sympathy without causing pain.
- Said distance hug, looking much like the picture at top, is also used in psych hospitals, where physical contact is prohibited, but patients wish to show support for one another nonetheless.
This can also be done on webcam in online relationships or when partners are apart.