The main muscles that are working the most when you run are your quads, hip flexors, hamstrings, glutes and calf muscles.
Running also works and tones your core muscles (these are basically the muscles of your torso, from the pelvis to the shoulder, and include the abdominal, lower back, buttock, pelvis and hip muscles), to keep your spine aligned as you run. Running is an effective core workout, thanks to the consistent spinal rotation that occurs when your arms and legs come forward with each stride. Run on uneven surfaces to further challenge them.
Strong core muscles also increase stability and help maintain good form throughout. It is therefore worth working on your core away from running as a strong core can help the main running muscles work at their optimum efficiency.
Running also requires you to swing your arms which works and tones the biceps and triceps.
Endurance running helps aid the development and efficiency of slow twitch muscle fibres, so their aerobic capabilities improve and they become more resistant to fatigue.
A schematic showing some of the skeletal muscles of man. The schematic was based on an image in the book “The human body” by Linda Gamlin
(Licensed under Wikimedia Commons https://commons.wikimedia.org/wiki/File:Skeletal_muscles_homo_sapiens. JPG)