You can run any appliance in your camper as you would in your house. However, the overall electric usage in your camper is limited.
More than likely, you have a 30 amp limit on your camper. If you plug into a 20 amp or a 15 amp circuit (plug) in your house, you can only use the 20 or 15 amps and then the breaker in your house will pop and you'll to reset it and turn something off in your camper.
Your lights are powered by your battery, which is 12 volts (DC). Your furnace is powered by your battery, which is 12 volts (DC current, not AC household). There's a converter in your camper that will convert your 120 volt AC power from the house to 12 volt DC for your camper that runs the DC electric items. That converter also uses power also. So if the ONLY thing you had on in your camper is the electric heater, you will be under the total 120 volt - 20 or 15 amp provided by your house. But remember, everything else you plug is is taking some of that 20 or 15 amp. Once the overall usage exceeds your circuit breaker in the house .... POP! Turn something off in the camper.
Running an electric heater should be no problems, perhaps a television also, a your cell phone charger plugged in ... you'll be fine. If you start running your microwave, or turn on the electric in your water heater, or start using a hair blower, you'll go "POP"!