If you have received a Notice to Appear or other summons to a removal or deportation hearing, it is critical that you make every effort to attend on the stated date, and at the stated time. In fact, you should plan to arrive early, in case parking is difficult, and because you are likely to have to pass through a security check.
As the Immigration Court Practice Manual warns (Section 4.8), any delay or absence from either a master calendar (scheduling hearing) or your individual merits hearing may result in the hearing being held "in absentia" -- that is, without your presence.
Since you would not be able to defend yourself and present any claims you might have for relief, the judge is likely to order you deported. Once issued, an order of deportation is very difficult to undo. You cannot appeal it, but would have to file a Motion to Reopen.
The first question is why you feel you can't make the hearing. If it is your first hearing, and you are worried that you do not yet have an attorney, you should go anyway. Volunteer attorneys are often provided to help you through this first hearing, as described in Will a Lawyer Be Provided For My Deportation Hearing?.
If all else fails, you can represent yourself "pro se" at the master calendar hearing. This is not recommended, but you can ask that the judge reschedule you to another master calendar date (grant a continuance). And no matter what happens, it's better than not showing up at the hearing and receiving an order of deportation.
If you really can't make it to the hearing, however, you will have to hire an attorney on your own, and have him or her file what is known as a "motion for continuance." This is not something where you can simply call up the court and say, "I need a different date."
The motion to continue the hearing to a later date should be submitted in writing, according to any applicable court deadlines, and contain the reasons why you need a continuance, as well as evidence to back it up, such as a doctor's letter saying you will not be released from the hospital until a later date.
You will need a good reason why you cannot attend -- missing work, for example, is not likely to be enough for the judge to agree to the continuance. Your hearing is part of a law enforcement effort, and the authorities don't want people who have no right to remain in the U.S. to use reschedule requests as a way to stretch out their stay.
You are allowed to state your (and your attorney's) preferred dates for a rescheduled hearing. Realize, however, that the Immigration Judge has discretion to simply choose a new date according to the court's schedule.
Merely filing a motion to continue does not mean you can ignore any scheduled hearing. You will have to wait until the motion is granted. If that doesn't happen before the scheduled date, you and your attorney will be expected to show up at the hearing as originally scheduled.
It is also possible to request what is known as a "Motion to Advance" the hearing date -- in other words, to set it for an earlier time. This, too, requires a good reason, such as an impending surgery.