DMARC - Demarcation point. In the USA this is defined by the FCC (47 C.F.R. Part 68) as a network interface device (NID). It often contains a lightning arrest and a test jack usually RJ-11. This can be inside or outside the structure, even on the edge of a property.
MPOE - Main Point of Entry. (Sometimes just POE) I've also seen this rendered as Minimum Point of Entry. Is is the point at which a data cable enters a property. This can be anywhere inside or outside the actual structure, but is usually immediately inside a building. circuits must be extended from this point into your IDF.
As to that IDF closet (also sometimes MDF.) that stands for Main Distribution Facility and Intermediate Distribution Facility. These can be the same location and is a room often as small as a closet which contains all that aforementioned telecommunications equipment. I usually call it a telco closet. When used in the technical sense, the MDF is the primary telco room, and usually the DMARC and the IDF is a secondary location in the same facility.
What's the difference again? The short answer is that the MPOE is the "physical" point at which the provider's lines cross into the customer's building (or sometimes across a property line). The DMARC is the "electrical" point at which the provider's lines connect to the customers equipment. This difference is important because the hardware and support on one side belong to one party and vice versa.
How did this happen? Think back to the Carterfone decision. Prior to the break up in 1984, AT&T owned everything. They had a total monopoly from handset to handset and everything in between. They owned the local loop, the DMARC, MPOE, cabes, poles, on-premise equipment, off-premise equipment, data centers, switches— everything. (Similar situations existed with small local phone companies as well) But even before the break up of AT&T the use of third party equipment created a need to delineate the portion of the network which was owned by the customer and the portion owned by ma bell.