Two separate meanings are in play here. The first, defined by the Oxford English Dictionary as “A cause of laughter; a joke,” was emblazoned onto my brain by the very sharp HBO-BBC comedy series “Extras.” In it Ricky Gervais plays a mediocre actor who finds himself starring in a witless sitcom, wearing a curly black wig and oversized glasses; his signature catchphrase, endlessly derided, is “Are you ‘avin’ a laff?”
The OED (characteristically, IMHO) makes no distinction, as it probably should do, between the plural and singular forms. That is, it is a standard Americanism to refer to “(having) some laughs,” while the singular, “a laugh,” is rare here. But not unheard of: a New York Times headline on November 27 was “Schmekel, a Band Born as a Laugh.”
The other meaning–“an amusing or entertaining person”–presents a more interesting case. The OED says this is “now chiefly Brit.” That means it’s an unusual example of an Americanism that became a Britishism and is now on the verge of NOOB-dom. For the American origin, the OED cites Kodak Magazine from 1921 (“Eight finished acts were presented, including‥Sam Kellman, Hebrew comedian, who was a laugh from start to finish”) and Of Mice and Men, 1937: “Old Susy’s a laugh—always crackin’ jokes.” It might also have mentioned Lorenz Hart’s 1940 lyric to “Bewitched, Bothered and Bewildered,” which notes with a characteristic double entendre, “he’s a laugh, but I like it, because the laugh’s on me.”
Fast forward to the 2000s, and the word has a strongly British identity. In the film “Shaun of the Dead,” Shaun says of Ed, “I like having him around, he’s a laugh.” And in Sue Townsend’s 2006 novel Queen Camilla, a character refers to “Prince Harry, who were a right laugh but were a proper ginga.”
The phrases laugh riot and laugh a minute have, of course, long been popular in the U.S., and they make searching for a-person-as-a-laugh challenging. However, my college daughter Maria reports that in her circles, it’s a comer. And I’ve found a few outcroppings on the internet, for example, this September 28, 2011, post from the blog thepartywhip.com:
But I just don’t see that friendly banter working with some of the goons who occupy the Tea Party wing of the GOP. “Hey guys, this is Ted. He’s convinced that the president was born in Kenya and is waging a steady effort to take away his gun, but believe me, he’s a laugh!”
notoneoffbritishisms.com 
I don’t know, Ben. It definitely goes both ways with the vocab. Lately I’ve heard several British people saying “truck.” The walls of Jericho are falling.
But still no “lorry” sightings here!
Ben, you don’t mention the chief British usage of “Are you having a laugh?”. It pretty much exclusively means “Are you taking the piss?” That’s to say, I couldn’t imagine it ever being used outside of that context.
Mr. L., I bow to your native speaker’s knowledge of the phrase. All I know from it is “Extras” and the OED, which doesn’t mention the “taking the piss” bit.
There is a definite misunderstanding of the of “Are you ‘avin’ a laff” – à la Gervais, which has its roots, I believe, in Estuary English.
He does not use it in the OED sense described.
Some one who “is a laugh” (or a scream) is someone who is funny ha; ha.- fun to be with, a joker. Something that “is a laugh” could be a play, film, book for example, or an activity – bungee jumping, for example – which is fun, often to the point of being a bit daring or even naughty like decapitating all the garden gnomes in the neighbourhood. Doing either of these two activities could be described as “having a laugh” and those caught in the act of Gnomicide might protest to the arresting police officer, “We woz only ‘avin’ a laff”.
However, “Having a laugh” as in Gervais’s “are you ‘avin’ a laff” means either, as another commentator has pointed out, a polite form of saying “Are you taking the piss?” or maybe an incredulous, “Are you being serious?”.
“Having a laugh” might be considered equivalent to both “joking” and “joking around”.
But also (particularly in the phrases “You’re having a laugh/Are you having a laugh?”) there is the connotation of someone being ridiculous or pushing their luck (c.f. “taking the piss”) – there’s an implicit “… at my expense”, i.e, “Are you mocking me, because you’re surely not serious?”
As a native English-speaker of the estuary variety, “Are you ‘aving a larf” also carries with it the potential for violence if the answer is incorrect.
There is another use of “a laugh” in colloquial BrE that hasn’t been mentioned: it can be used to mean “a good time”, e.g. “That party was a proper laugh!”