I have a vague notion of what a filtered diagram is, and I believe that a filtered colimit is a colimit of a filtered diagram. Here we see a definition that uses some language I do not understand:
https://ncatlab.org/nlab/show/compact+object
They say
Definition 2.1. Let C be a locally small category that admits filtered colimits.
What does it mean to say that C "admits" filtered colimits?
What does it mean to say that a category "admits" something?
Very generally in math, some structure X "admits" Y if Y exists for the structure. So a category that admits filtered colimits is a category in which every filtered diagram has a colimit.
(Personally, I find the usage of "admits" in this case to be a bit odd. Usually "admits" refers to the existence of some additional choice of structure that is not unique or canonical. For instance, you might say an abelian group admits a ring structure if there exists a multiplication operation which together with the addition operation of the group makes it a ring. Such an operation is usually very far from being uniquely determined. On the other hand, colimits, when they exist, are unique up to unique isomorphism. So, they aren't really something extra which the category "admits"; I would just say the category has filtered colimits.)