A maintenance request is a formal way of asking your landlord to repair something that you can't do on your own. Maintenance requests should always be written and dated so it helps both you and your landlord keep track of the request and make sure everything is taken care of.
There are some things that need a maintenance request and should be fixed by a landlord and there are some things you can take care of on your own. The main thing to keep in mind is if something requires the help of a professional, such as a plumber, locksmith or HVAC repairman, you shouldn't try to take care of it on your own, even if it seems like an easy task. This will help you avoid liability in case the problem ends up being worse than you thought.
If there's ever an urgent issue, such as a flood or fire, you shouldn't worry about filing a written maintenance request, but instead should contact your landlord and the appropriate emergency service immediately.