A mileage run is an unofficial term adopted by many veteran frequent fliers that distinguishes flights flown for the sole purpose of earning the most “miles” or points in one’s frequent flier program.
Most fliers who do mileage runs do so on the weekend, when they have time off of work. Mileage runs are done in earnest towards the end of the year, as fliers scramble to gain the required “miles” needed to ensure the airline status they want for the next year.
A representation of this would be a frequent traveler who, in November, realizes they need 9000 more SkyMiles on Delta so that they can continue as a Gold Medallion for the next full year. The traveler then decides to do a mileage run on one weekend, to earn the miles that they need, while paying the least amount of money. Leaving Friday after work, they do a trip, from ATL-BOS, while making a stop in LHR, SAP, IAD, and CDG (bad example, but you get the point). After returning, the flier now has sufficient SkyMiles to ensure Gold Medallion status on Delta for the next year.
Its counterpart in the hospitality industry is referred to as a mattress run.
One more note, many fliers who do mileage runs may never even leave the airport; if done on the weekend, a roundtrip flight with 2 or 3 layovers each way, which are typical in mileage runs, can leave very little time to leave the airport’s secured area with time enough to get back in for the next flight out.