Missed April 15th deadline to accept grad school offer. What can I do?
You have nothing to lose by reaching out -- the sooner the better. The best person would be a professor who recruited you (wants you in their group) or informed you of the offer. But I would also reach out to:


Any administrator who contacted you
Any other professors you are interested in working for
Whichever faculty member in the department is in charge of graduate admissions


The email doesn't need to be long, but it is important. Just apologize and say that it was a mistake, but you had intended to accept the offer. Unless they are looking for an excuse to retract the offer, there is a good chance they still want you. They may see missing this deadline as a bad sign; however, being proative and apologizing for your mistake will make it more likely that they forgive it. Follow up and reach out to new people if you don't get a response.



P.S. Bryan Krause makes an important point in the comments about programs that have wait lists:


  The biggest issue is that any program with a wait list is likely to go to those applicants as soon as possible, which is the fairest thing to those applicants. They may not have a funding slot for you if someone from that list accepts, and they may feel they should give them priority over you[.]