A further technical point: it is the offer of funding (TA, RA, or fellowship) that expires on April 15. The admissions offer is usually still valid (at my institution, you have two years to choose to matriculate, and the offer still stands even if you have declined it). This is a moot point for most, since most students cannot afford to pay for graduate school, but it may be relevant if you discover that the school you chose to attend is not a good match.
In terms of the exact time of day, if it is not specified in the offer letter, it is probably midnight (your time, the school's time, UT, or some other time zone...). For practical purposes, however, like admissionprof said, I also recommend considering it as 5pm in the time zone of the institution making the offer.
However, I would like to point out that not all departments view variations of 1-2 students as negligible. Admissionprof's case is for a Physics department. In Astronomy, our target class size may be only a few students, so 1-2 is a large variation. In our department, we usually make more offers than we expect to accept (over-commits), but as April 15 nears we are trying to hit an exact target number. We have made offers on, or after, April 15. If you are on a waitlist, you should be in contact with the departments and express your continued (or not) interest in their program. There is no exact science to resolve the grid-lock/domino effect of competing waitlist/offer scenarios, but direct communication of the situation is the most palliative measure.
Finally, if you are holding more than 2 offers at this time, please be a responsible citizen and decline the programs that you are no longer interested in attending. You do not need to have made a final decision in order to notify the other schools. Directors of Graduate Study, Chairs of Admissions Committees, and anxious waitlisted students are all checking their in-boxes regularly today.