In the Sherlock series, I noticed this peculiar thing which has been hinted at, many times. Sherlock is so weird. Almost no one like him and vice-versa but what makes John Watson so different? How come Sherlock starts to like him so much? Sherlock has hardly shown any emotion during the show except at those very few times when John is in danger. If it is not inappropriate I want to explore the chemistry between these two characters. What makes Watson and Sherlock such good friends?
Sherlock likes John because he appreciates him unlike others who call him a freak and ask him to piss off. In the episode A study in Pink, when John and Sherlock are in the cab the following conversation happens:
John provides Sherlock a generalized opinion as to how a normal person would react to the given situation. During his best man speech Sherlock says,
John also is his blogger. In the episode The Great Game Sherlock does mention, "I will be lost without my blogger". Apart from that, their affection does seem to become vivid in The Sign of Three where Sherlock confronts to John.
For John liking Sherlock, here's a transcript reference from His Last Vow
To sum it up Sherlock and John need each other, whether they want to or not. Even when they are frustrated at each other, they always drift back together.
Source for transcript: Ariane DeVere
Sherlock likes Watson because Watson is useful, he accommodates Sherlock's habits and plays his part in things. For example Watson has no issue with Sherlock not speaking or eating for days and then suddenly demanding his full attention. In the books more than the series Watson is also clever, insightful and resourceful. Holmes notes in stories where he narrates himself that Watson continually plays down his role in their partnership.
Watson likes Holmes because he provides some physical excitement for Watson as well as intellectual challenges of the cases - he also likes to help people and these cases allow him to do that. Watson looking after Holmes when he drugs himself out of his mind due to boredom is another way he can help. The books are slightly less extreme than the series in this regard, Holmes has odd habits but Watson is also, primarily, a loner who appreciates alone time and does not want a housemate who constantly bugs him.