1. Literally, to continue moving, swinging, bending, tilting, etc., slowly in one direction and then back in the opposite direction. The structure swayed back and forth, threatening to topple at any given minute. Everyone in the crowd was swaying back and forth in time with the song.

2. By extension, to oscillate between two opposing positions, decisions, inclinations, points of view, etc. Public opinion has swayed back and forth on this issue for decades. You can't keep swaying back and forth over this—you need to make a decision and commit to it!