Cool Brian
Cool Brian: Wholesome Mentors For Growing Boys is a mentorship corporation with different "Brians" working for them. Most Brians are college students or have just gotten out of college. They get paid to hang out with "growing boys." The main idea is that all the Cool Brians have to be responsible so it rubs off on the boys they spend time with. Any irresponsible activity would probably get them fired.
Plot
Rowley's parents hired a Cool Brian employee for Rowley (who was rather old for it, but due to his childish behavior, he was very happy with it). Rowley was seen playing catch in his front yard with an employee. On his regular employee's day off, he played with a different "Brian", but, as Greg Heffley noted, Rowley didn't seem to notice.
Rowley's relationship with the company presumably came to an end after he started to stay away from older teenagers, after believing he could "catch puberty" from them, after watching a puberty video.
Relationships
Rowley Jefferson
Brian is, as Greg puts it, sort of a "big brother for hire" whose job is to become a boy's "best friend," but Brian is mainly doing his job and does not really have strong feelings for Rowley. Rowley doesn't seem to notice, however. At one point in time, an entirely different "Brian" was seen playing catch with Rowley, but Rowley didn't care.
Greg Heffley
Greg is actually quite jealous of Brian, because he was at first Rowley's "mentor" until Brian came along. But Greg's supposed "mentoring" was mainly teaching Rowley childish pranks, which is probably part of the reason Rowley's family hired "Cool" Brian. He also wanted to bet money that the guy's real name wasn't even Brian.