Meaning of the Halting problem
We first choose an alphabet of symbols that our Turing machines can read and write on the tapes. Typically we have three symbols: $0$, $1$, and "empty". A word is a finite sequence of symbols.

If $u$ and $v$ are words we can form a new word $\langle u, v \rangle$ which represents the two words put together (this requires some coding so that we can tell where one words stops and the other begins).

A Turing machine can be be described by a word. 

Like in all decidability problems, $H_{TM}$ is a set of words. More precisely, $H_{TM}$ contains all those words of the form $\langle M,w \rangle$ where $M$ is a Turing machine, $w$ is any word, and $M$ halts when we run it with input $w$.