I want to make a door similar to this:
What kind of wood would you recommend and what would be the best way to build it?
I am guessing it would need to be reinforced from the back?
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Generally-speaking, you can use whatever type of wood and finish appeal to you.
Depending on your application, you may want the door to be either lighter or heavier. If the door is closing off a movie room, you may want a more solid, thicker, heavier door to better absorb and deaden the sound. Of course, if you really want to block sound, you'll probably also want to seal off gaps around the doorway between the wall and the door.
If the door is simply closing off the kitchen from the hallway and you will need to frequently open and close it but preventing sound from passing through is not a major concern, you may prefer some more lightweight type of construction.
The metal straps in your picture may be critical to the door's structure, but it's highly likely they are only aesthetic. Ideally you would drill oval horizontal holes in the faux metal straps, cover the holes with washers, and avoid overtightening the screws. This will allow the wood to expand and contract across the width of the door (across the grain, as pictured) as the humidity changes with the seasons.
You can assemble the door panel as you would any typical panel or tabletop, then install the faux metal straps for visual enhancement.
If you use solid wood, quartersawn lumber will be less prone to warping. If you prefer the appearance of flatsawn/plainsawn lumber, be sure to alternate the end grain as discussed in https://woodworking.stackexchange.com/a/241/49.
You could build the door using plywood or MDF (or even MDF-core plywood), using veneers that give the appearance of a solid wood panel, if preferred. If you use plywood, be sure to use a high-quality, many-ply type of plywood such as Baltic birch.
As keshlam noted in a comment on your question, you could also build a hollow-core door, more generally described as a torsion box. This involves building a flat internal skeleton, with thinner outer "skins."
That door is held together by the metal strips across the face.
another which can be all wood would have the 'Z' on one side. You can use any wood, pine would be fairly light and easy to move, but easy to ding up. Oak would look nice, but be a little heavier, but a good rolling system shouldn't make it a big deal.
ETA: when making one of these it is recommended to do a 'tongue and groove' style. Put a groove on all the boards, then glue in a 'tongue' (spline) on only one. Fit them together and then put the braces on. By not glue both sides you let the wood move without twisting the door. [The complete illustrated guide to furniture & cabinet construction by Andy Rae] Just read that chapter this weekend end. Source
Focusing more on the wood itself in that picture my first instinct for stability and ease of assembly was to use some hardwood flooring.
Since hardwood flooring is already tongue and groove it will make for a very strong door with minimal effort. The metal would then be used to keep the wood together aside from glue possibly.
The couple of downsides would be the weight and that wood usually has a concave recess on the bottom of the boards. Fixing both of those should not be hard as you could plane or cut with the saw to make a nice flat surface. Something similar would need to be done about the outside tongue and groove, but again, good pass on the table saw will true that.
Since you don't need much it is usually easy to come up with small amounts used. I have seen second hand sites offer that up regu