Assuming you mean a day in terms of an Earth day in both cases, the phenomenon I believe you're referring to is called time dilation, which affects the speed at which time is experienced by one body relative to another. This rate is affected by velocity and gravity, each of which can cause the bending of spacetime that results in the discrepancy between objects. Both gravity and velocity slow one's time as they increase.
The situation you describe is probably not naturally occurring, however is not outside the realm of reality. As referenced by Wiki from Calder's book Magic Universe, Calder claims that an acceleration of a constant 1G (what we feel on Earth all the time) would result in the affects of time dilation allowing you to traverse the entirety of the known universe in a single human lifetime (for the traveler). Conversely, during your trip, the rest of the universe would be aging 'normally'. There are plenty of technical problems with the feasibility of achieving this, but is conceptually possible.
I'm not particularly qualified to explain the equations behind it, so I'll leave that to our more capable members. Coincidentally, time dilation is the excuse I give when asked why I walk so quickly.