I have a 400mm square piece of 18mm MDF. I need to take a 300mm by 300mm square of depth 10mm out of the middle of the piece of wood. Currently the only tool I really have is a Dremel, so I don't plan on doing it with that. Is it possible to cut out something like this with a router? If so which bit would I need?
It's exactly the job for a router. I'd recommend using something like a 1/2" (~13mm) cutter, with 2 straight flutes as a minimum - if you can get an upcut spiral bit or a 3-flute cutter, or even a cutter specifically for MDF then that may be better. A carbide-tipped cutter will fare better/last longer but is more expensive than a basic steel bit. MDF is pretty abrasive on cutting tools anyway, so cheaper may be a better way to go.
Basically just take it slowly with the feed speed and you should be alright. Maybe practice on a small scrap piece first if you can. Routing MDF is easier than routing timber because you don't have to pay attention to grain direction. You'll want some good extraction and/or to do it outside with a good dust mask as the dust from MDF machining is nasty.
You might also want to construct yourself some sort of rudimentary guide or just use some planks of wood clamped across the width of the workpiece, moving it along to support the router base - you could run with the router base directly on the surface of the MDF but you might run into trouble with the router tilting if you're not careful.
From your description WhatEvil's answer is spot on. Router is a great solution for this project given the materials suggested and desired outcome
You don't need a router if that is what you are trying to make if you are willing to change your materials....
You would have to buy different thicknesses for this to work but another idea would be to buy two smaller thicknesses of MDF and use something simple like a scroll saw, jig saw or coping saw to cut out your shape out of one and then laminate that to the other piece.
Cut your square out of a piece of 9mm, for instance, and glue that to another 9mm. You don't get your 10mm in that case but perhaps that is acceptable? you do however get the original 18mm of your first piece!