The log was a simple float the Challenger scientists used to measure surface current. From an anchored boat, sailors threw the log attached to a rope into the water. Scientists noted the direction the log was moved by the current. After a period of time passed, they measured the length of rope pulled overboard by the float and divided this distance by the time, to get the speed of the surface current.
The current drag measured currents below the surface. The weighted drag hung below a surface float so it could drift along in deeper currents. Scientists calculated the velocity of the drag's surface float, and taking into account the surface current measured with a log, they calculated the velocity of the deeper current.
Modern current meters have various designs. Basically they either move with the current as drifters, or are stationary and hold one position as the water moves past.