Q:

What's the name of this puzzle?

In this puzzle there are 25 squares, and you have to connect the pink square with the purple square.
Here are the rules:

You walk on every square to arrive the purple square before arriving.
You can't walk diagonnally
You aren't allowed to step only once on every cell

What's the name of this puzzle?

A:

I would call it impossible.
Imagine the board is a checkerboard, so the moves allow you to step from a black cell to a white one and vice versa.
Let's position the board that the pink cell is above a white checkerboard cell, so there are 13 white cells and 12 black cells in the setup given.
If you are allowed to step only once on every cell (this is not clearly stated in the problem statement), then you cannot do this, as there is no path through 25 (or any other odd-numbered) cells, which starts at a white cell but ends at a black one.

