Baud rate is the number of symbols that can be transmitted over a line in a second. This is similar to bit rate except that each symbol usually consists of more than 1 bit. So in a system with 3 bit symbols operating at 1000 baud will have a 3000 bit/s bit rate.
Bandwidth is the difference between the upper and lower frequencies of a given piece of spectrum and is measured in Hz. This is essentially the amount of space available to transmit data through the air or over a wire. Think of cable TV; multiple channels are able to be carried over a single wire because each channel uses its own frequency range, and the width of that frequency range is the channel's bandwidth.
Channel capacity is the maximum data rate that can be carried over a certain medium given several factors including the amount of bandwidth available. The greater the bandwidth, the greater the channel capacity.
Because of the direct relation between bandwidth and channel capacity, channel capacity and throughput are often referred to as bandwidth in the computing world.