Bandwidth, when used in computing, refers to the rate at which digital data can be transferred, typically measured in bits-per-second (bps – all lower case). Thus a a gigabit ethernet connection might be said to have a maximum bandwidth of 1 gigabit per second, or 1gbps.
Bandwidth is also colloquially used to refer to an individual’s capacity for information consumption. “I don’t have the bandwidth for that” might be the response of someone who’s been asked to do too many things at once, or who is simply flooded with more information than they can process.
Technically, the term is a misnomer, since there’s no real “width” associated with computer bit speeds or information consumption. The term originates in radio, where a transmission said to be at a particular frequency actually spreads into frequencies immediately nearby as well. The frequency and the “width” of that spread defines the “band” that the transmission occupies.
The usages are related since in both cases, higher or larger bandwidth implies a higher or larger capacity to carry information.
For more excruciating detail, see bandwidth on Wikipedia.