The KATRIN experiment is a next-generation direct neutrino mass experiment with a sensitivity of 0.2 eV (90% C.L.) to the effective mass of the electron neutrino. It measures the tritium β-decay spectrum close to its endpoint with a spectrometer based on the MAC-E filter technique. The β-decay electrons are guided by a magnetic field that operates in the mT range in the central spectrometer volume; it is fine-tuned by a large-volume air coil system surrounding the spectrometer vessel. The purpose of the system is to provide optimal transmission properties for signal electrons and to achieve efficient magnetic shielding against background. In this paper we describe the technical design of the air coil system, including its mechanical and electrical properties. We outline the importance of its versatile operation modes in background investigation and suppression techniques. We compare magnetic field measurements in the inner spectrometer volume during system commissioning with corresponding simulations, which allows to verify the system's functionality in fine-tuning the magnetic field configuration. This is of major importance for a ... mehrsuccessful neutrino mass measurement at KATRIN.