As you surmised, Edison power uses +120V and -120V rails with a neutral, so both 120V and 240V connections are possible. Wait, this just in: Tesla won and everything has gone AC, so the + and - signs are only accurate half the time :)
Also, there's now a requirement for a safety ground wire. This wire does nothing except handle fault current, and has saved thousands of lives. It cannot be combined with another wire (that's been tried to death).
Anyway, you're an electronic person, so fasten your seat belt. Mains has a lot of weird rules, that from an EE perspective seem insane, but if you bear with it, it will eventually come together to make perfect sense.
What you have there
What you're looking at is not genius. You are looking at the bog-standard 40A or 50A electrical feed for an electric range, which is in almost every home. I bet if you look next to it, you'll find a gas line because it was converted to gas, leaving the heavy cable unused. The cable will be 8 or 6 AWG Cu, or possibly 6 AWG Al. (Don't panic about Al, we have ways of dealing with that)
The cable will have 3 or 4 wires, but more likely, 3. One of them may be bare, but if there are only 3 wires, there is no ground wire, which is a serious issue. We'll come back to that.
Here's what some fools do
Seeing as they have no electric range, and want more kitchen receps, they grab onto those heavy wires and extend them to ordinary receps. This would be just fine if several things are true:
- The breaker is changed to 15A or 20A
- There is a ground wire (or it's GFCI protected)
- There is a neutral wire
Typically, they shortcut one of these. The classic is failing to change the breaker to an appropriate value.
Now you know what you have.
Types of 240V range connection
Unfortunately there are several types of range connection out there in the wild. 3-wire connections are a serious problem, because they do not bring out a ground wire; *but even worse, the official Code and UL-approved recommendation is to abuse neutral as ground. The logic is these connections don't fail very often. But if they do, the oven chassis is energized with 120V -- which violates a principle of mains electrical design, which is that a simple failure should not create a hazardous condition. So I do not recommend this, even though it's "leeegal".
A 4-wire connection: Black, Red, White, and green/bare ground. This is the modern, safe connection type. You are sitting pretty if you have this.
A 3-wire connection: Black, Red, White. Hot L1, Hot L2 and Neutral respectively. If you have this, you have 3 options: 1) use as-is; 2) fit GFCI; or 3) retrofit ground.
A 3-wire connection: Black, Anything, stranded-mesh bare. This use SE cable as the wiring, using two insulated conductors for hots, and neutral is a bunch of bare strands which encircle the other conductors. This is legal despite the bare neutral, because of an exception in Code for using SE (Service Entrance) wire for the range. You can 1) use as-is, 2) fit GFCI, 3) retrofit ground and continue the mesh as neutral, or 4) re-task the bare wire to be ground, and fit a 240V-only range there.
A 3-wire connection: Black, Anything, solid bare wire. Uh-oh. They cheated when they installed it. That is "/2 w/ ground" cable. By the time that came along, "/3 w/ ground" cable also existed, and you were supposed to use that, since it was available. So that is illegal, and cannot be continued in service as a hot-hot-neutral connection. Your only options here are to a) run a whole new cable, or b) re-task the bare wire back to ground and use a 240V-only range.
3 loose wires in non-flexible metal conduit all the way back to the panel. You're in luck. The metal conduit is an allowed ground path, so you simply ground the range to the metal junction box.
3 loose wires in plastic conduit. Fish a fourth ground wire into that conduit, and you are done. There might be an issue with conduit fill, but a #10 bare ground wire doesn't add much fill.
A 3-conductor cable, in a sheath, in a conduit. See the above restrictions on cable. It's legal to run cable in conduit if the conduit interior is at least 138% of the cable's widest dimension (the cable, treated as round, can't fill more than 53%). However it's a bad idea because it's hard to pull through. Retrofitting a ground into a conduit with cable means there are now 2 wires in the conduit, and fill rules require 30% fill, so the conduit must be 185% of the cable's width (not likely). In that case, run the ground outside the conduit, or pull the cable out and run individual wires.
You cannot retrofit a neutral wire. All related conductors must run in the same cable or conduit, for several reasons relating to AC. Ground doesn't count as a conductor because only fault current moves on the ground wire, so it can take a different route if need be.
Any white wires used as a hot must be re-marked (sleeved) by wrapping tape or by shrink tubing. The tape must be a legal hot color (NOT green yellow-green white or gray).
If you are stuck with a 3-wire connection, I recommend at least that you put a GFCI breaker on it. (in these sizes, GFCIs come as breakers, not receptacles). Pricey, but they will protect you from the aforementioned scenario by tripping the breaker to prevent shock. You must remove the neutral-ground jumper strap on the range. If an oven receptacle is involved, you change the recep and cord to NEMA 14-50 (4-prong grounded).
The other option for a 3-wire connection is to retrofit a ground wire. It needs to be a 10 AWG ground wire, and it needs to go back to the service panel, or any point with a 10 AWG ground wire back to the panel, e.g. water heater or A/C or the grounding electrode system. You then separate neutral and ground on the range.
The last option, in certain cases, is to employ a 240V-only range and oven. This is a bit of a trick to find, since the oven light is typically 120V. (That, and the clock and controls, is about the only thing that is 120V). That's because any common incandescent enjoys the heat, but it'll destroy a CFL or LED. A subsidiary to this option is hack your range to take 240V-only (no neutral) by installing an appropriately sized supply transformer inside it; i.e. large enough to power the oven light.