Node c-commands node if and only if
Consider the tree above, which is in line with X-Bar Theory. By the above definition of c-command, we can, for example, say that X0 c-commands ZP and everything that ZP dominates. This is so, because X0 does not dominate ZP and ZP does not dominate X0, and furthermore, the first branching node that dominates X0 (which is X') also dominates ZP. ZP, in turn, c-commands X0. Moreover, X' c-commands YP. The lower segment of XP, finally, c-commands WP.
The first passage in the definition of c-command is supposed to guarantee that a node does not c-command a consituent that the first node dominates. If there were only the second passage, this could very well happen since you can go follow a line of from a c-commanding node to the first branching node and then take the same route back - since what the definition says is that you just have to go up to the first branching node and then take a look at the constituents this branching node dominates.