Telephone lines have power that is separate from your usual home electrical service, so when your power goes out a telephone will often still work.
I have both cable and dsl, and my dsl modem is on a large UPS battery backup. When power goes out I can, and have for a whole day, use the Internet over the dsl connection with a laptop and several large UPS batteries while all other power is out.
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Telephones with cords do work without electricity, which is why every home should have at least one corded phone that will operate in the case of a power outage. Cordless phones generally require electricity.
The original telephone was purely an electrical device as opposed to the modern telephone, which, although it depends heavily on electricity, is more nearly an "electronic" device.
The original telephone was composed of a battery, a simple microphone, some wire, and a simple speaker. These were connected in a circuit in which the microphone varied the amount of electrical current drawn from the battery, passed through the wire, and applied to the speaker. As a result, speech at the microphone was reproduced by the speaker. Since the signal representing the speech was carried from the microphone to the speaker by the electrical current flowing in the wire, electricity itself was fundamental to the operation of the telephone.
As the telephone system was developed, more functions were added, such as ringing and dialing systems. These depended fundamentally on electricity as well.
In the modern age, telephone systems have been converted over to electronic systems, basically using transistors for the most part at some level, but still entirely dependent on electricity.
Along the way, radio systems using radio waves, and more recently, fiber optic systems using light, have become a part of the telephone system we know. While these systems do not pass their signals via electricity as such, the electronic systems that make them work require electricity to function.
This is why we say that the telephone system will not work without electricity.
There is, however, at least one example of a "telephone" that works without any electricity at all. That is the old two cans and a length of string. For most purposes the electrical telephone system works a lot better!