Modular Decomposition and Clique-width
you will find an introductory text on clique-width (cwd for short) here: Upper bounds to the clique-width of graphs (B. Courcelle and S. Olariu, DAM 101). You can find more recent results in this survey: Recent developments on graphs of bounded clique-width (M. Kaminski, V. Lozin, M. Milanic, DAM 157(12): 2747-2761 (2009))

Cwd is a complexity measure based on graph operations that generalise word concatenation. Infinite countable graphs can have bounded cwd. You will say that a set (possibly infinite) of graphs (finite or countable) has bounded cwd if there exists a constant k such that any graph in this set has cwd at most k. For instance, complete graphs have cwd 2, distance hereditary graphs cwd at most 3, ... 

1) The link between cwd and modular-dec is the following: cwd(G) = max {cwd(H) | H prime in the modular dec of G}. Hence, you can say that cwd generalises the fact that "prime graphs has bounded size". You can have graphs with prime-graphs of unbounded size but with bounded cwd.

2) if the size of prime-graphs is bounded, the cwd is bounded. The results in the paper you cite says that any problem expressible in MSOL can be solved efficiently in graph classes of bounded cwd. This set of problems include many NP-complete problems: clique-number, stable number, 3-colorability, ...

Some algorithmic aspects of  modular dec are studied here "A survey of the algorithmic aspects of modular decomposition" (M. Habib and C. Paul, Computer Science Review 4(1): 41-59 (2010))