Middle Back

The primary target of the rowing machine are the muscles of the middle back. The motion of leaning back to pull the handle toward the chest causes the trapezius muscle (lower, middle back) to engage as the primary workers.

Lower Back

By changing the rowing technique, you can further engage the muscles of the lower-middle back, the latissimus dorsi. To target these muscles, do not lean at all while performing the exercise, keeping the back straight while pulling the handle towards your chest.

Arm Muscles

The rowing machine engages the biceps as the handle is being pulled toward the chest. Alternatively, it can engage the triceps if you use a wide-grip handle instead of the standard v-grip.

Leg Muscles

While you should keep the legs slightly bent throughout the exercise, the rowing machine does engage both the hamstring and the gluteal (i.e., backside) muscles. The body naturally will use these muscles to stabilize you during the row.

Other Minor Muscles

Other muscles engaged during this exercise as minor stabilizers include the deltoid (shoulder muscle) and many of the muscles surrounding the spine. These muscles aid in posture and reduce the chance of lower back pain.

Source:

Bodybuilding.com - The Row

Resource:

Interactive Diagram of Back Muscles