A crossover frequency is usually defined as the frequency at which a system crosses unity gain (gain = 1 = 0dB). A corner frequency (which I think is the term more appropriate to the context of this part of the lecture) is the frequency at which an "abrupt" change in slope occurs. These occur at the poles of the system.
For the system G(s)=1/(s+1), there is a corner frequency at ω=1 rad/s. The terminology probably comes from the fact that the asymptotes for ω<<ωc and ω>>ωc form a corner. It so happens that at the corner frequency, the actual gain response is 3dB lower than the horizonal asymptote, so that is how we define our corner frequencies.
The notion of bandwidth conveys the maximum frequency at which the gain response of the system is maintained. This is defined more strictly as the frequency at which the system gain goes below 3dB of unity (-3dB). Therefore, above this frequency the input signal is attenuated by a factor greater than 3dB. Note that for the first order system shown above, this value occurs at 1 rad/s, which is also the corner frequency.