What are HP and MP in this context?
Both HP and MP are decision problems, namely sets of instances that you can think of as descriptive strings. For example,

HP is a collection of pairs $M, x$ where $M$ is the description of a Turing Machine (think of a listing) and $x$ is an input to that machine. An instance is such a pair and a pair is in the set HP if the machine described by $M$ eventually halts when started on input $x$,

and, similarly

MP is a collection of pairs $M, x$ as above and an instance of MP is in the set MP if the machine described by $M$ eventually halts in an accept state when started on input $x$.



In simple terms, HP is about programs and inputs and answers "Does this program halt on this input?" and MP answers "Does this program accept this input?" 

The octothorpe (the # symbol) is just a separator so we know where $M$ ends and $x$ starts.