My house was built in 1949 and some of my outlets look like this guy while others run updated Romex into plastic boxes. With the wires coming through the conduit in the plaster, are these grounded outlets? I was going to replace the outlets and want to make sure I understand what is going on with these before I do. Thanks!
Back in 1949 when your house was built, the code of the time allowed the metal shielding of the BX cable to be used as a ground. Over time these connections degrade from corrosion, and the grounding could eventually fail. Oftentimes, when a home is renovated to use non-metallic wiring (NM), the ground to existing BX cable can also become disconnected. This is likely the case with your house.
There still is a possibility that the ground is intact, but it is unlikely. You can test this with an inexpensive outlet tester like pictured below. These typically run around $10, and are a good investment. In your case, you would probably see that this outlet has an open ground. Make sure that the outlet is properly secured to the box before you test it.
If you find that the outlet is not grounded, you should avoid using it until you can have that part of your home rewired.
This is a picture postcard application for a GFCI outlet. Attach it to the black and white wires without ground. That is code legal for this type of situation, but the outlet should be labeled "no equipment grounding conductor". At that point you can plug 3-prong appliances into it. The GFCI provides much of the protection of the ground. This doesn't satisfy every appliance, but it is, at least, not unsafe.
I an skeptical that this is actually grounded because I see both conductors disappearing into a 1/2" hole, and with all the debris on the bottom of the box, I don't see the normal hardware I'd expect to see from the conduit joiner or the BX clamp. An unguarded knockout hole is a real problem. It may look ok now, but when you put the outlet back in and tighten it down, it could pinch and damage the conductor.
This is not 1949 work. The outlet is grounded and worse, has those vile "back stabs". That introduces the real possibility that the wire-run is a retrofit also, in which case, maybe it is practical to pull new cable the same way.
Those 3-light outlet testers are neat, but keep in mind they can trip GFCI's. The hot-ground yellow lamp is, by nature, a ground fault. (so is the neutral-ground red lamp, but the GFCI is unlikely to detect that one.)