A mode in HCI is a context where user actions (keypresses, mouse clicks etc.) are treated in a specific way. That is, the same action may have a different meaning depending on the mode.
Modes are known to cause problems as you may press a key expecting it to do one thing thinking the system is in a particular mode, but something unexpected happens because in fact it is in another mode.
However, modes are often essential as there are only a finite number of input options (keys and mouse actions), but often a large number of thinsg you would like them to do.
The crucial thing is to ensure, as far as possible, that mode are apparent to the user via visual, aural or other forms of feedback. Furthermore, it is important to ensure that mode errors, because they are common, do not lead to catastrophic consequences.
Examples: If you have a (old fashioned!) mobile phone and press "999" you would expect to be about to ring the police (in the UK), but if you do the same thing while editing an SMS message you would get 'y' (assuming multi-tap input) - there is a phone mode and a calculator mode In a graphics editor click and drag on the camvas might mean draw a line, circle, or rectangle depending on what tool is selected - that is a mode for each tool. This may be more major, in the example in the video, we see that Google Docs drawing app hass a 'normal' mode where clicking over a menu selects it, but when the polylne tool is selected, clicking adds another line segment – here a 'normal' mode and a polyline mode.
For more: see HCI book case study – "Excel mode error"