Okay, so I'm going to try and explain things. Sorry if I over-explain, but judging from your comment of not knowing what the battery disconnect switch does I am assuming you know almost nothing about your RV power system.
Most RVs are designed to be able to work with or without shore power. Because of this, in general, the water pump, lights, vent fans, and furnace blower run on 12 VDC (the tongue jack and slides also run on 12 VDC). The fridge, if it's an absorption fridge, can use 12 VDC to light a propane flame which cools the fridge, or it can use 120 VAC if you have shore power to save propane. The water heater is usually similar. Wall outlets, microwave, TV, AC, etc all use 120 VAC. What this boils down to is that by using batteries, run everything in the camper except TV/AC/wall outlets, allowing you to use most electric systems from batteries while camping at sites with no electric. We do this quite often.
Now, this means you have two separate electric systems on board. One is 12 VDC, the other is 120 VAC. Your coffee pot, microwave, TV, and fireplace all run off 120 VAC, whereas your lights and furnace run off 12 VDC. So if your furnace is shutting down and lights are going out but NOT the TV, coffee pot, and microwave, then you have a problem with the 12 VDC system, and not your 120 VAC system. Got it?
Okay, so the 12 VDC system. This consists of a battery (or batteries), a fused distribution block, a 120 VAC to 12 VDC converter (what I think is bad), a battery disconnect switch and your DC loads, such as your furnace and lights. The battery only powers the furnace and lights IF you don't have shore power. If you have shore power, then the converter steps in, taking the 120 VAC shore power and converting it down to 12 VDC. Think of it like an oversized phone or laptop charger. When the converter is powered, the battery no longer has to power the loads, and in fact the converter will charge a dead battery in addition to running your lights and furnace.
Batteries don't like sitting around at much less than a full charge for long periods, they will sulfate and be permanently ruined. Because things in your camper will always use some 12 VDC power even when shut off (shut as your audio system), this means if your camper sits the battery will get drained, eventually (or quickly) ruining it. That's the purpose of the battery disconnect switch. It separates the loads in your camper from the battery, keeping it from discharging when your not using your camper. In normal operation, you should turn off the battery disconnect when you park your camper for storage if you don't leave it plugged in, and turn on the battery disconnect when you hook up to tow your camper.
So, long explanations done, let's dig in and give you a few things to test. I believe your converter is either not powered on or not working. Hopefully these tests figure that out and get you on your way, or let you know what you'll need to have fixed.
Battery Disconnect: If this is on (battery indicators show "full"), does the furnace and lights still cycle? Sounds like it does, but I just want to check. This should be on when camping, off when not camping if the camper isn't plugged in.
Power: Does the converter have power? Check your circuit breaker panel. Usually the converter is on its own breaker and labeled as such. Be sure the breaker is on and not off or tripped. Try flipping the breaker off and then on again.
Output: If you have a multi-meter (sounds like you're camping so you probably don't), you'll want to measure the converter output. There's usually large metal blocks near the automotive fuses in the DC side of your distribution panel. One will have a really thick red wire, the other a really thick white wire. Measure the DC voltage between these blocks. It should be 13.6 volts or so (up to 14.4) if your converter is running. Anything 12 or below and the converter is bad.
Fuses: The converter can't power any loads or charge the battery if the output fuses are blown. These are often multiple fuses parallel, right next to each other. They are the biggest fuses in there. I have two 40 A fuses. Pull these fuses and make sure they are not blown.
