If an offense has a mandatory-minimum sentence, that means that the law requires a trial court to, at a minimum, impose a certain defined sentence. The trial court can go beyond the mandatory-minimum sentence, but it must at least impose the minimum sentence set forth in the law.
For example, consider the offense of Resisting or Obstructing a Peace Officer. The statute defining the offense, 720 ILCS 5/31-1, requires either 100 hours of community service work or 48 hours in jail. The Unified Code of Corrections also states that this offense is not eligible for judicial supervision. As such, at a minimum, the trial court must convict the defendant of the offense (supervision is not a conviction) and impose either the community service work or send the defendant to jail. The Court could go beyond this minimum, for example, by sentencing the defendant to a month of jail and 150 hours of community service work, however, the Court could not go less than the mandatory minimum.