> Does it mean simply lack of pattern? But that doesn't seem to be the case, at least visually.

Aperiodic tiling means that the whole thing doesn't repeat. If you overlay a copy of the tiling and move it around, you'll never find it match perfectly everywhere except for the one position you cloned it as. A grid of squares is periodic, you can translate it one unit to the side and it's the exact same, everywhere.

This is of course a different kind of periodic than is meant with The Periodic Table.