Modem (from modulator-demodulator) is a device that modulates an analog carrier signal to encode digital information, and also demodulates such a carrier signal to decode the transmitted information. The goal is to produce a signal that can be transmitted easily and decoded to reproduce the original digital data. Modems can be used over any means of transmitting analog signals, from driven diodes to radio.
The most familiar example is a voiceband modem that turns the digital 1s and 0s of a personal computer into sounds that can be transmitted over the telephone lines of Plain Old Telephone Systems (POTS), and once received on the other side, converts those 1s and 0s back into a form used by a USB, Ethernet, serial, or network connection. Modems are generally classified by the amount of data they can send in a given time, normally measured in bits per second, or "bps". They can also be classified by Baud, the number of times the modem changes its signal state per second.
Baud is not the modem's speed in bit/s, but in symbols/s. The baud rate varies, depending on the modulation technique used. Original Bell 103 modems used a modulation technique that saw a change in state 300 times per second. They transmitted 1 bit for every baud, and so a 300 bit/s modem was also a 300-baud modem. However, casual computerists confused the two. A 300 bit/s modem is the only modem whose bit rate matches the baud rate. A 2400 bit/s modem changes state 600 times per second, but due to the fact that it transmits 4 bits for each baud, 2400 bits are transmitted by 600 baud, or changes in states.
Faster modems are used by Internet users every day, notably cable modems and ADSL modems. In telecommunications, "wide band radio modems" transmit repeating frames of data at very high data rates over microwave radio links. Narrow band radio modem is used for low data rate up to 19.2k mainly for private radio networks. Some microwave modems transmit more than a hundred million bits per second. Optical modems transmit data over optical fibers. Most intercontinental data links now use optical modems transmitting over undersea optical fibers. Optical modems routinely have data rates in excess of a billion (1x109) bits per second. One kilobit per second (kbit/s or kb/s or kbps) as used in this article means 1000 bits per second and not 1024 bits per second. For example, a 56k modem can transfer data at up to 56,000 bits (7kB) per second over the phone line.

Narrowband/phone-line dialup modems

28.8 kbit/s serial port modem from Motorola
A standard modem of today contains two functional parts: an analog section for generating the signals and operating the phone, and a digital section for setup and control. This functionality is actually incorporated into a single chip, but the division remains in theory. In operation the modem can be in one of two "modes", data mode in which data is sent to and from the computer over the phone lines, and command mode in which the modem listens to the data from the computer for commands, and carries them out. A typical session consists of powering up the modem (often inside the computer itself) which automatically assumes command mode, then sending it the command for dialing a number. After the connection is established to the remote modem, the modem automatically goes into data mode, and the user can send and receive data. When the user is finished, the escape sequence, "+++" followed by a pause of about a second, is sent to the modem to return it to command mode, and the command ATH to hang up the phone is sent.
The commands themselves are typically from the Hayes command set, although that term is somewhat misleading. The original Hayes commands were useful for 300 bit/s operation only, and then extended for their 1200 bit/s modems. Faster speeds required new commands, leading to a proliferation of command sets in the early 1990s. Things became considerably more standardized in the second half of the 1990s, when most modems were built from one of a very small number of "chip sets". We call this the Hayes command set even today, although it has three or four times the numbers of commands as the actual standard.