It looks pretty straight forward. The sentence is talking about two kinds of "relations", partial order relations, and equivalence relations. It is saying that the property of "transitivity" is a key property in each.
If your question is about the meanings of those words, a "relation", in mathematics, on sets is any set of "ordered pairs" of members of those sets. We can think of the fact that (a, b) is in that set as meaning that a and b are "related" in this way. In particular a relation is said to be an "order" relation if and only if, whenever (a, b) and (b, c) are pairs in the relation, so is (a, c).
A relation is an "equivalence" (we sometimes use the notation "a~b" to indicate that (a, b) is in the set of ordered pairs) relation if and only if it satisfies 3 conditions:
The "reflexive property": for any member, a, of the base set, we have a~ a (that the pair (a, a) is in the set of ordered pairs).
The "symmetric property": for any two members, a and b, of the base set we have that if a~b (that if (a, b)is in the set of ordered pairs) then b~ a (that (b, a) is in the set of ordered pairs.)
The "transitive property": For any three members, a, b, and c, of the base set we have that if a~b (that if (a, b) is in the set of ordered pairs) and if b~ c (if (b, c) is in the set of ordered pairs) then a~c (that (a, c) is in the set of ordered pairs.
In either case, the order relation or the equivalence relation, "transitivity" is an important property.
