Bandwidth, the rate at which data can be passed through a connection.
This terminology has been used to identify different forms of electronic communications, some of which have been misuses by some such as web hosting companies, whom often refer to it when identifying the amount of data that one has for use on a monthly, or other basis.
Causes of this have been over time the confusion people had with the identifying of the rate of transfer, and that in the past these terms were not widely known to the whole population. These terms include such things as Kbps, Mbps and Gbps. Since data limits are measured in similar terms as KB, MB and GB, people often identified this as bandwidth because of the similarities, and it stuck.
Mbps is Megabits per second, this is different to Megabytes. A Megabit is 1,000,000 bits of data. a Megabyte is 1,024,000 bytes of actual data. So this means that to download 1 MB of data in one second you would need a speed of roughly 8 Mbps.
The reason for this is that there are 8 bits per byte. So the display of Mbps as the speeds that you can achieve on an internet plan is always more impressive than MB/s. If you look at the speed of the internet and thinking it is running slow, check your plan and see how many Mbps you are supposed to be able to achieve as a maximum, then divide that by 8 to get a rough idea of the maximum you will be able to achieve in MB/s or whatever the alternate denotion for the plan is.
In the future, hopefully the world will be able to identify once again, and be able to understand the differences and how to avoid mistaking a rate of transfer and a size of a chunk of data.