About the pairing

In Super Junior, there are 105 possible pairings. Among them, there's this one that became more and more popular over the years : Kim Jongwoon and Cho Kyuhyun.
There are so many interesting things about it but I'm not going to list them all here, because I just wanted to make a short intro.
So, what can we say about this pairing ? without being biased

A lot of different names :
International : KyuSung (규성) / YeKyu (예규)
Japanese : GyuYe (ギュイェ)
Korean : HyunYe (현예)
Chinese : KyuWoon (圭云)
Other name (mostly used by spanish speakers) : YeHyun (예현)

Some observations :
-there's a larger number of Clouds (Yesung's fans) who ship KyuSung, compared to Gamers (Kyuhyun's fans).
-there's a lot of KyuSung shippers in Japan...like, a lot.


Rivals ?
In singing ? Maybe. A lot of people believe that. But I don't think it's something serious. If it was the case, then why would Yesung want Kyuhyun to win Immortal Song just like he did himself ?

Skinship ?
Contrary to some beliefs, they do a lot of skinship. Not as much as others (EunSiHae, anyone ?), but they do have a lot of moments like hand holdings and philtrum touchings and various physical interactions.

Real friendship ?
Kyuhyun said they are close, and Yesung once felt like they were true friends. Plus, he said that Kyuhyun always stays to comfort him when he's in a bad mood, and he's the first one to be by Yesung's side when he feels tired. So yes, I personally think they're good friends.
But some people may think they are not, probably because Kyuhyun just can't stop teasing Yesung about everything. Which is pretty stupid, in my opinion. Because firstly, Yesung thinks it's cute and, most of the time, isn't annoyed by all this (though he can get angry or violent sometimes and I'm not surprised), he even said Kyuhyun is someone he can't dislike although he plays pranks on him all the time. And secondly, Kyuhyun surely does this by pure affection. Teasing aside, he can be really sweet towards Yesung.

People could say this "friendship" is just for fun, it could have been all made up...but there are some things that look too genuine to be considered otherwise.
[I mean, you can't force yourself to cry when you're going to lose someone, right ?]

This is not their whole relationship of course, but I think you have the basics now.

If you want to know more about KyuSung :
You can go read this and this (made by Alice) :D
And also, there are two lists that could help you : 20 reasons to ship KyuSung and Top 30 Best KyuSung Moments.

If you want to know more about the way people see this pairing :
Why KyuSung ? : all the comments are really interesting.