Broadband is a synonym of "Wideband". These are engineering terms.
In engineering terms, any medium is Wideband if it can carry a "big band" of frequencies.
For example, in older days cables used for TV could carry only a few MHz of frequencies, that limited number of channels that could be transmitted over those cables. If you transmit more channels there will be a crosstalk and voice/video of one channel might mix into another.
Cables that are used today can carry hundreds of channels simultaneously.
A typical TV channel requires 6 MHz band. So any transmission medium that can carry frequencies from 500 to 600 MHz can carry 16 channels at max.
Another medium that can carry frequencies from 500 MHz to 1000 KHz (1 GHz) can carry much more channels.
broadband means wide "band". Band is a range of frequencies used for communication. Some communication methods such as FM use a few KHz above and below the frequency assigned to them. For example, 98.3 FM uses a few KHz above and below 98.3MHz to transmit their signals. Broadband uses the same principle except the the "range" of frequencies are quite wide. Thats why its called _broad_ band. It has no implications whatsoever on the "bandwidth" or "download" speeds that we get. The speeds could be 64Kbps and still it can be termed as "broadband".