A mandatory minimum means exactly what it sounds like. It is a floor on sentencing that the judge is legally required to impose regardless of any mitigating circumstances, personal history, or preference for leniency.
When a charge carries a mandatory minimum, the judge loses the discretion they would normally have to craft a sentence based on the individual circumstances of the case. The legislature has effectively made that decision for them by writing the minimum into the statute. A judge who personally believes a lighter sentence is more appropriate has no authority to impose one if the mandatory minimum applies. The law removes that option entirely.
What this means practically is that the sentence imposed will be at least the mandatory minimum amount. The judge can go higher if aggravating factors justify it, but they cannot go lower. For drug offenses in particular, federal mandatory minimums have produced sentences that many judges themselves have publicly criticized as excessive, but their hands are tied by the statute.
There are a limited number of ways around a mandatory minimum. The most significant is substantial assistance to the government, meaning cooperating with prosecutors and providing information that leads to the conviction of others. A substantial assistance motion filed by the prosecution can allow a judge to sentence below the mandatory minimum, but that motion has to come from the government, not the defense. The defendant cannot simply offer to cooperate and expect relief. Prosecutors have to agree it is worth filing.
Beyond cooperation, some jurisdictions have safety valve provisions for certain drug offenses that allow first-time nonviolent offenders to be sentenced below the mandatory minimum under specific qualifying conditions. Whether that applies depends entirely on the charge and the jurisdiction.