to give rest or to take rest: to relieve someone from toil and refresh them, or to pause and be refreshed oneself
The word pictures relief from work. It is built from a verb meaning "to make something stop," so at heart it is about bringing labor to a halt long enough for a person to recover. Depending on who receives the rest, it points in two directions: giving rest to another, or taking rest for oneself.
When someone gives the rest, the word can mean handing a tired person real relief. Jesus' promise "I will give you rest" is the deepest form of that. On an everyday scale it is what friends do when they lift each other's spirits: Paul says the believers "refreshed my spirit," and he asks Philemon to "refresh my heart." The related noun (ἀνάπαυσις) names the rest itself, the state or the resting place.
When a person takes the rest, it ranges from a nap or a short break to the settled repose of those who have finished their life's work. Revelation promises the faithful dead that they will "rest from their labors." And in one place the word describes the Spirit of God coming to settle and remain on a believer, echoing the old promise that the Spirit would rest on the Messiah (the Hebrew there is נוח). A close cousin, καταπαύω, carries the heavier idea of God's own settled, Sabbath-like rest.