What kind of move am I looking for?

The goal is to find the best move, or sequence of moves, in the given position.

You do not always have to find a checkmate! Just find the best move.

Usually, this move will lead to a superior position, either by a forced sequence of moves which leads to checkmate, or (more commonly) wins substantial material.

By "substantial material" we mean usually winning at least the exchange. More commonly the winning move will net a whole piece (bishop/knight/rook) and sometimes will win the queen. Occasionally, the material will be only a single pawn--this usually happens in endgame situations where the extra pawn will likely decide the game.

The first move is not always the most difficult move to see. Sometimes, the initial move in the sequence is somewhat obvious, but the real solution to the position lies in the follow-up moves. In order to solve our puzzles, you must see enough moves to demonstrate that the initial move is correct. Simply guessing the first move, without understanding why it works, is not solving the puzzle.

Sometimes we will present a position where the player who is to move is in a nearly hopeless situation. In these positions you are expected to look for a way to draw the game instead of win it. We don't tell you that you are looking for a draw; you are expected to figure this out by the nature of the position.

I agree with you, with the caveat that Nd3 is a forcing move but not a forced variation. That's exactly the difference I was exploring.

My point is more about habit than correctness. I grant you that Nd3 has a very small tree, and you can compute all the lines easily. Nevertheless, it's a good habit to play a move like Ra1 instead. You can't get bit by a zwischenzug when you give check, and it's much harder to overlook a reply. I furthermore believe that exploring forced moves by habit is important to improve your tactical vision, and makes your solutions are easier to construct.