When you slog, you toil at something, working hard and often slowly to get a difficult job done. You might slog with your shovel through the pile of snow in your driveway.
A worker might slog through a pile of papers on her desk, or slog long and hard on a construction crew. In either case, the job requires persistence and determination. Another way to slog is to walk with difficulty, the way someone might slog along a muddy road to the gas station after getting a flat tire. The original meaning of slog was "hit hard," possibly as a variation of slug.