Absent a local requirment, NM (the plastic sheathed cable) is fine. It is also less expensive initially and requires less labor for installation.

Armored (could be AC or MC, but most likely AC since there is not a separate ground conductor) and often referred to as 'BX' (the first cable type with spiral armor, but long outdated) does not add much protection.
MC is often used in commercial work, especially with steel studs. NM can be used but requires bushing of every hole through a stud the cable runs through.
The installation rules for nailing plates are the same for both.

Are you by any chance located in the Chicago area? The reason I ask is that's one area that requires conduit for residential uses that would not require conduit in most other parts of the country. There may be other areas with this unusual requirement as well.

It's worth checking for kinky local code requirements before proceeding. Otherwise, NM-B oughta work fine EXCEPT where the wire runs are on unfinished basement walls or similar places that need additional protection against physical damage (e.g., not protected by finished walls and not run along overhead joists where they are out of normal reach).

Metal conduit (EMT) is far more labor intensive to run than cable but some areas require it (notably Chicago and it's suburbs). Because it costs more, most builders don't use it unless local code requires it. You should call your local building code department before starting on your basement. A bunch of Romex coming out of the panel in an area that requires conduit is going to create problems down the road if you ever sell your home.

Bending conduit neatly takes some practice, so if you're going to do the work yourself, I'd suggest buying some extra pipe. :-)

Is conduit "better"? Well...that a subject of some debate. The biggest advantage I see is that it makes replacing wires or adding circuits a easier because you can get back to the service panel with relative ease. It also offers greater physical protection than cable (especially non-metallic cable [e.g., Romex]). Although proper installation of cable will offer adequate physical protection. EMT will also keep the rats from eating your wires but if you don't have rats, that's probably not important. :-)

The conduit itself functions as the equipment ground, that's why there isn't a separate ground wire. That's not an absolute, there are some exceptions. Pool and spa circuits come to mind.

Wow, thank you. As it turns out, we are in a suburb of Chicago, so I bet that's it. I'll check with our building inspector when he calls back about the permits.

Fun times ahead. Are there any tips or tools for bending the conduit neatly?

Thanks

"Are there any tips or tools for bending the conduit neatly?"

Get the correct size hickey and a good handle.

If you are around Chicago the odds are good that conduit is required although some places do allow NM-B (Joliet comes to mind).

Bending conduit takes some practice and while there are some websites with information (try Google, I don't have a specific recommendation), you should expect to make a few mistakes. You'll need a conduit bender, one for each size of pipe you'll be working with (1/2" and maybe 3/4"). They're available at Menards or Home Depot for about $35. Ideal, Klein, and G-B are the brands you'll find and brand probably doesn't make a big difference. You'll need a hacksaw to cut it (or you can use a jig saw or reciprocating saw with a metal cutting blade). You'll also need to clean up the cut ends so you don't damage the wires. Klein makes conduit fitting pliers, Klein and Ideal make conduit fitting screwdrivers, and I've seen guys use an ordinary pair of channel lock type pliers. All depends on what you are comfortable with and/or want to spend.

Remember you install the conduit first and then pull your wires.

Here is a link that might be useful: How not to install conduit