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In electronics, Bandwidth is used to measure electronic communication. It is defined as the width of the range of the frequencies that an electronic signal uses. Bandwidth is expressed in terms of the difference between the signal having highest-frequency and the signal having the lowest-frequency. In computer networks, bandwidth is often used as a term for the data transfer rate. More easily, the amount of data that is carried or passed from one point to another in a network, in a given time period (usually a second). 
Many systems work by means of vibrations, or oscillations (Vibrations that goes back and forth at a regular rate). Each complete cycle of "back and forth" is called, simply enough, a cycle. The number of cycles per second of a system is its frequency. Frequency is measured in cycles per second, usually called "Hertz", and abbreviated "Hz".
Most systems do not operate at just a single frequency. They operate at many different frequencies. As an example, sound travels through vibrations. It has at least one frequency, and usually many different frequencies. People can hear sound frequencies as low as about 20 Hz, and as high as about 20,000 Hz. A band of frequencies is a continuous range of frequencies; the band of frequencies people can hear is from 20 Hz to 20,000 Hz.
To define, bandwidth is the width of such a frequency band, that is, the highest frequency minus the lowest frequency. In the hearing example, the bandwidth of a person's ears is about 20,000 Hz - 20 Hz = 19,980 Hz.
Bandwidth is often applied to the electromagnetic spectrum (e.g., radio waves, light waves and X-rays). Such waves are vibrations (oscillations) of electric and magnetic fields. To use a radio example, the lowest United States AM radio channel covers the band of frequencies from 535,000 Hz to 545,000 Hz. It therefore has a bandwidth of 10,000 Hz (545,000 - 535,000). All United States AM radio broadcasting stations have this bandwidth (though the location of each band is distinct). The lowest United States FM radio channel (on the other hand) covers the band from 88,000,000 Hz (88 MHz) to 88,200,000 Hz (88.2 MHz). It therefore has a bandwidth of 200,000 Hz. (Notice that the width of an FM band is 20 times the width of an AM band.)
The term "bandwidth" has been misused in the field of digital data communication. It is often incorrectly used to mean "data carrying capacity". However, there is no such thing as "digital bandwidth". The proper term for the data carrying capacity of a communication channel is channel capacity.
It is true that, in general, the channel capacity of a system increases with the bandwidth used for communication. However, many other effects are also important. Therefore, in many (if not most) real systems, the channel capacity is not easily related to the channel bandwidth.
Sometimes, the word "broadband" is used to mean "high-speed", especially in for high speed internet connections. "Broadband" means "wide band", and suggests high-speed. However, the term is not clear; "high-speed" is more clear. Typically, a dial-up telephone connection is thought to be low-speed, at less than 56,000 bit/s (bits per second). High-speed is usually 200,000 bit/s or faster. DSL (Digital Subscriber Line) or Cable modem connections are usually high-speed.