I came across the phrase in a book about an old ‘great house’ that has been turned into a sort of hotel. If it matters, it was an interior wall that had been wallpapered. A character is walking down a hallway and notices something about a section of the wall and thinks something like ‘probably has the Damp. Shame.’
From context, it’s more than simply being wet, it’s something damaging, bad. Water condensation leading to mildew/mold, perhaps?
Damp, or rising damp, is essentially just the UK term for “water damage”.
It’s almost certainly referring to rising damp, as that is the most common use of the word in this building context.
You comment may be misunderstood. Rising damp is dampness coming up from ground level as the name implies. It is not just any kind of water damage, which would of course cause dampness - such as a leaking roof.
“The damp” sounds archaic. Nowadays we’d just say “it’s got rising damp” or “it’s got damp”.
Agreed. There was supposed to be more words in there. Thanks for the clarification.
Ah, very interesting! Thanks for the link, RNATB.
Now, why haven’t I ever heard about that kind of damage here in US houses? I’ve heard about dry rot, and termites, and black molds…
Okay, according to the linked article, England had laws about water-proof barriers by 1875, and knew the source of the problem before that. And there really aren’t all that many house in America older than that, so likely it’s a pretty rare thing.
Ignorance fought. 
Most houses in the UK are made of brick, which would wick up water out of the ground. In modern houses a plastic damp-proof membrane (‘damp course’) is added at about the second course above ground level (and certainly below the level of the floorboard joists).
In older properties, the damp course was formed by painting bitumen, or laying bitumen-impregnated felt on the top of the bricks at the appropriate point during construction, and this eventually breaks down and becomes permeable, allowing damp to rise.
In very old properties, there was no damp course at all - damp walls would have just been the norm.
In some properties, subsidence or a change in the soil surface level results in the dampcourse now being below ground level, allowing moisture to enter the brickwork above it.
Some of this is, as you note, associated with the fact that we have a lot of very old properties still standing, but the problem exists even in some built since WWII - building standards have not always been very rigorously observed.
Also, it rains here quite a lot.
There were other forms of damp-proofing in older houses:-
*There are a number of alternatives to the modern dpc-on-a-roll, and these are sometimes found on older properties, particularly those built prior to the Second World War. The most commonly encountered alternatives are the brick damp course and the slate dpc. The brick dpc uses a clay brick with a very low absorption rate as a barrier to rising damp. The slate versions also rely on the impermeability of the slate to stop the ‘wicking’ of water via capillary action.
Good point. My last house (built in the 1930s) had a slate dampcourse - mine was OK, but many of my neighbours with similar properties had experienced failure (the remedy for which was injection of resin into the brickwork).
I think the common failure modes for slate DPC are: spalling when moisture enters through the exposed edges, then freezes, or accidental bridging in the case of cladding or rendering.
The DPCs I have seen on houses seem to be made of those blue “Staffordshire engineering” bricks.
The corollary to Mangetout’s post, of course, is that brick construction is pretty rare in the US, and concrete does not absorb moisture as readily as fired brick.