If you use first offender, the record will show that you were charged with the offense, but it will not be entered as a "conviction" provided that you complete all the terms of your probation while on first offender. Thus, in a manner of speaking, it will "come off your record." However, first offender, as the name implies, may only be used once. In the event that you're charged with a felony later on down the road (which is what first offender is generally reserved for), you won't be able to use it. The question of whether or not to use first offender hinges on (1) the nature of the charge and (2) how certain you are that you will not need first offender in the future. I'd weigh the options carefully.
While the case is pending and the sentence is being served a criminal history will show that you were arrested for the offense but that there is no conviction. If you successfully complete the first offender sentence then either you or your attorney needs to ensure to that an order discharging the first offender sentence is signed by the court. Once that is done it is possible to have your record expunged, which would clear even the arrest from the sort of criminal history that can be obtained by employers and private parties. It would still be possible for law enforcement agencies and some other government agencies to obtain histories that would show the arrest and first offender disposition for the offense. It may be possible for you to fill out the expungement forms on your own but consulting with an attorney is generally a good idea.
Also, as the previous poster said you can only use your first offender once, and if you use it for a misdemeanor then it would not be possible to use it again if you were arrested for a felony. A final item to consider is that if you don't successfully complete your sentence/probation on the charge where you are using the first offender then they can revoke your probation and re-sentence you up to the maximum sentence for the initial offense. So if you do use the first offender, be sure you comply with the terms of probation, get the order discharging your sentence, and fill out the expungement paperwork.
One other thing to consider is whether your offense is one that can be resolved by diversion. Many counties have diversion programs for offenses like shoplifting and minor possession of alcohol. If successfully completed, the state will dismiss the charges. There are some counties that require the assistance of an attorney for an accused to be accepted into diversion, so it may be worth your while to consult with a local attorney to see if there is a program and if you qualify. Then you wouldn't have to use first offender.