The term dominate refers to a relation in trees. A given node in a tree dominates all those other nodes below it. If you can trace a line from a given node N1 to another node N2 moving only downwards (never upwards), then N1 dominates N2. Note that when there are two basic relations in trees, dominance and precedence. A given node N1 can dominate another node N2, and/or it can precede it. N1 precedes N2 if N1 appears to the left of N2. Look at the tree here: https://en.wikipedia.org/wiki/C-command. Read the definition of c-command and the explanation. Examine the tree. I think the concepts are well explained there.
The c-command concept (or just command) stems from the 1960s. There are numerous versions of the concept: command, c-command, m-command, etc. The particular version used depends on the grammarian. In general, the fact that the concept changes and is used variably bears witness to problems with the general idea.
Please be aware also that while the X-bar theory and c-command are concepts that are widely taught in linguistics and syntax courses, they are controversial. Many approaches to syntax reject these notions. In the place of c-command, they tend to take the syntacitic functions (=grammatical relations) to be basic and ranked among each other. The subject is rank ahead of the the first object, and the first object is ranked ahead of the second object, and the second object is ranked ahead of a prepositional object. Many phenomena of syntax are then addressed in terms of this ranking of functions.
Here are some further links that might be helpful: