Given a model of a system and an objective, the model-checking problem asks whether the model satisfies the objective. In this case, the model is a standard graph, and the objective is Büchi: given a set of target vertices $T\subseteq V$, determine whether or not there is a path that visits the set $T$ an infinite amount of times. Furthermore, in the conjunctive problem, you are given multiple Büchi objectives (i.e. multiple target sets $T_i$) and you are to determine whether or not there is a path where all of the Büchi objectives are satisfied (i.e. whether or not an infinite path visits all target sets $T_i$ an infinite amount of times).