The divorce process can involve all kinds of emergency situations. Maybe your spouse is threatening to take your child out of state, or the mortgage is about to foreclose unless somebody pays it. You can file motions any time you need the court to address issues while your divorce is pending. A motion to expedite consideration asks the judge for special dispensation so you don't have to wait as long as it might ordinarily take to get into court.
Filing and Service
Some states, such as Maryland and Alaska, require that you file one motion asking for whatever you want the court to order, and a separate motion asking to expedite the proceedings. The judge will then decide when to schedule the hearing – either granting your request and holding it relatively immediately, or denying the motion to expedite and scheduling the hearing at the regular time, usually in a few weeks. In other states, such as New Jersey, you can file one motion, asking in the submission that the hearing be held on short notice.
Likelihood of Success
The judge may not agree with you that your situation is an emergency warranting expedited consideration. Generally, courts will only grant expedited motion requests if irreparable harm will come to you, your property or your children if the matter is not heard right away. You can't make such a request just to hustle things along -- or if you do, the judge is not likely to grant it.