Depends on whether you asked for and received first offender status when you went to court on the charge. If you did ask for first offender treatment, and if you completed probation successfully without any violations, then your probation officer should present documentation to the court for the court to sign discharging you of the offense. Be warned though -- often times the paperwork does not get to the right place and the charge continues to appear on your record, and for that reason you might want to hire a lawyer to assist in making sure that the fact you have been discharged is properly recorded on your criminal history. If, on the other hand, you did not request first offender treatment (or if you did but later had a violation and had that status revoked, then you have a permanent conviction on your record that can only be removed through a pardon from the Governor.
It is generally impossible to have an arrest removed from your record once you have been convicted. If you were sentenced under the first offender act then your "conviction" may be marked "exonerated under first offender" or some such language. If you didn't get sentenced under first offender your record will likely reflect "convicted" for the rest of your life. A pardon may be your best option, but I believe that it's unlikely (if not impossible) to receive a pardon until many years after your sentence has ended. Whatever you do, don't violate your probation or get arrested again, because that would likely make a pardon impossible, not to mention you might have to go to prison!
If you have any doubts about your status you should hire an attorney, perhaps the one who represented you when the case went to court. Don't try to navigate the criminal justice system without a lawyer!