Cool Brian (also known as 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." Brian is, as Greg puts it, sort of a "big brother for hire" whose job is to become a boy's "best friend". 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. Ever since Rowley has been avoiding older kids, Greg hadn't seen Cool Brian for a while, Greg thinks that Rowley might've been avoiding "Cool Brian" too because he thinks they're contagious.
Initially, Rowley's parents hired a Cool Brian employee for Rowley to act as a sort of "big brother". While Rowley is rather old for this organization, due to his child-like nature, he legitimately enjoyed himself. At one point in time, an entirely different "Brian" was seen playing catch with Rowley, but Rowley didn't seem to care. Rowley's relationship with the company presumably came to an end after he began avoiding older teenagers, believing he could "catch puberty" from them.
Greg is actually quite jealous of Brian because he viewed himself as being 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.