Emergencies of the less serious type, like one where the tenant has been given timely notice and then finds it impossible to vacate the rental unit by the date specified, are the most common. See the landlord immediately if that happens. The rental unit may have been rented to another party. A tenant has no automatic right to remain in the rental unit beyond the last day of the rental period. The only way a tenant can "hold over" into the next rental period is with the permission of the landlord. Be sure both parties understand and agree upon what the cost is to "holdover." Remember, the last agreement was the rental of the unit for a specific period of time, for example, one month, and the tenant may not want to end up paying a full month's rent for one or two days' occupancy of the former housing.
The other type of emergency situation arises when the tenant's health or safety is threatened. The tenant may have to move to temporary housing and immediately notify the landlord. If repairs are not made in a reasonable period of time, the tenant may wish to consider deducting expenses from the rent payment or a lawsuit. Save receipts for expenses as it may be necessary to prove costs in a legal proceeding.