If something is flagging, it's worn out or weak. A flagging political campaign is running out of steam, losing the energy it needs to be successful.
If your career is flagging, it's languishing or fading — you might need to go back to school and start a new one. Flagging public support for a new school might spell the end of the project. A different kind of flagging is the sort you walk on. If your patio is made of flagging, it's paved in stones — flagstones, to be specific. The "weakened" meaning of flagging comes from an early meaning of flag, "to flap about loosely."