The first one is more likely. It means they have difficulty remembering things.
The second sentence would imply that the reason they forget words is because they keep remembering bad things. Perhaps whenever they try to learn words, they have bad dreams about some horrible English teacher, and this makes them forget the words they were learning. Possible, but I shouldn't think it happens very often.
"Memory" is the part of the brain that stores information. "Memories" are things that are stored in the memory. In computer terms, your "memory" is like your hard drive, and your "memories" are the things you've saved on your hard drive.
I myself have a very bad memory for names and faces. I have some very good memories of my trip to Vienna last year.