Cellular automata are well explained in their Wikipedia page. Most work on cellular automata is concentrated on systems where each cell can have a finite number of states; it is possible to consider systems with an infinite number of states, but this introduces all sorts of additional complications into the theory and it kind of defeats the whole point of cellular automata (which is, basically, that you can get very complex behaviour by chaining together simple finite systems with simple interactions; if you allow for infinite states then the complex behaviour isn't all that surprising).
The paper you link to considers quantum cellular automata, where the number of states of each cell is replaced by the Hilbert-space dimension of the state space associated with each cell; the paper is thus requiring that that dimension be finite.