Downtime is a variant rule of D&D fifth edition that acts as a sort of currency for side activities. Periods of downtime can vary in duration, but each downtime activity requires a certain number of days to complete before you gain any benefit, and at least 8 hours of each day must be spent on the downtime activity for the day to count.
Earning downtime, or DT for short, can be received from the completion of hunts, bounties, and quests. If you feel DT should be rewarded and hasn't, talk to your game master about it.