Philip Henslowe is the owner of the Rose, the theatre that puts on Shakespeare's comedies. He's in debt to Fennyman, and constantly riding Will to finish his play so he can make money off it. However, Henslowe wants a comedy.
He's a comic relief character who loves comedy. That's also, in his opinion, what the public wants:
"Love and a bit with a dog, that's what they want."
Henslowe's an idiot with bad teeth who thinks that Romeo and Ethel, the Pirate's Daughter is a "good title."
If they had hashtags, Shakespeare would be #teamtragedy and Henslowe would be #teamcomedy. Their bickering never amounts to about the same consequence as your average Twitter hashtag (i.e. it's forgotten in 140 minutes) but it does make us smile, especially the running gag about how every comedy should have a swordfight and a dog.
In a meta moment, a swordfight breaks out during rehearsals, and a dog joins the fray. Just as Shakespeare finds romance in real-life, Henslowe finds comedy in real life, too.