In most cases, if you were charged with DUI or DWI because you were caught driving while drunk, these are considered as misdemeanors. The charge can be enhanced or you can have multiple charges if you were also driving without a valid license or had other violations as well like speeding. The best part is that the Border Security, Economic Opportunity, and Immigration Modernization Act of 2013 (BSEOIM) forgives up to three misdemeanors and you will definitely be approved. Just make sure that you provide all the paperwork related to the court cases.
It can get tricky if you actually hurt someone because then it is a felony (laws vary by state, though), which generally means that you cannot be approved. So review the documents related to your cases and reconfirm if it was a misdemeanor. If it was a felony, and you are still here, you must consult with a lawyer before applying.