Arukone, or Numberlink, is a type of logic puzzle involving finding paths to connect numbers in a grid. The player has to pair up all the matching numbers on the grid with single continuous lines (or paths). The lines cannot branch off or cross over each other, and the numbers have to fall at the end of each line (i.e., not in the middle).
It's one of the Nikoli puzzles. The Nikoli website has four easy examples to play: