Veroboard is board that can be used for constructing simple electronic circuits. It is fiberglass board, with an array of holes drilled in it, and the holes linked by parallel copper track:
A is linked to B, and C is linked to D, but A/B is not linked to C/D
To use veroboard, you usually place parts in and solder them down, using the tracks to link to other parts. If using integrated circuits and the like, you often have to cut tracks between pins, by drilling through the copper strip.
Veroboard is good for very small circuits, but it's awkward to work with as your circuits get complicated. It can, however, be put to good use if prototyping a PCB: you can use it as a jig when drilling holes in your copper board, to make sure they are lined up right.