What is bit rate? How is it related to bandwidth?
As hinted by Mr Duval, the term "bandwidth" has been adopted by networking engineers to mean something similar to "transfer rate" (just as "wideband" in networking tends to mean "high speed"). You've tagged your question "digital communications", though, where the term is very precisely defined.

Bit is the unit of information in information theory. Intuitively, a bit is the information conveyed by the answer to a yes/no question, where each answer is equally likely.

In communications we want to transfer information from a transmitter (information source) to a receiver (information sink). The information transfer rate is measured in bits per second. Informally, you can think of this as the number of zeros and ones that a system can transmit per second.

However, it turns out that information is transmitted and received using physical, analog signals (voltages or electromagnetic (radio) radiation). These signals pass through a physical medium, called a "channel", that acts like a filter: only certain frequencies make it from the channel input to its output. So, the channel is said to have a bandwidth, just like any filter.

One of the main jobs of communications engineers is to transfer information as fast and reliably as possible over a channel of given bandwidth. Basically, the problem is how to design a physical signal that:


Fits over the channel bandwidth
Conveys bits at a high rate
Does so reliably (with few bit errors at the receiver)
Does so at feasible complexity at both transmitter and receiver.


The design of such signals is, in a broad sense, covered by the term "modulation".

I hope this is clear -- please comment if you need clarification.