What is the relationship between bandwidth and noise bandwidth
The noise bandwidth \$B_N\$ of a (linear time-invariant) system is defined as the bandwidth which an ideal filter with a rectangular frequency response would need to have to get the same noise power at the output, given that the input noise to both systems is identical and white. The ideal filter is usually assumed to have the same maximum gain as the system under consideration. From this definition, it follows that the noise bandwidth is given by

$$B_N=\frac{1}{H_{max}^2}\int_{0}^{\infty}|H(f)|^2\;df\tag{1}$$

where \$H(f)\$ is the system's frequency response and

$$H_{max}^2=\max_{f}|H(f)|^2$$

Definition (1) is valid irrespective of the specific characteristics of \$H(f)\$. So it is valid for lowpass systems as well as for bandpass systems, or other types of filters.

For a simple first order lowpass system (e.g. an \$RC\$ lowpass), we have

$$H(f)=\frac{1}{1+jf/f_c}$$
with \$f_c\$ the -3dB cut-off frequency. From (1) we get for the noise bandwidth

$$B_N=\int_{0}^{\infty}\frac{1}{1+(f/f_c)^2}df=
f_c\int_{0}^{\infty}\frac{1}{1+x^2}dx=
f_c\arctan(x)|_{0}^{\infty}=f_c\frac{\pi}{2}$$

In this case we see that the noise bandwidth is larger than the -3dB frequency by a factor of \$\pi/2\$.

For a second order system everything is similar, but slightly more complex. The integral gets a bit more involved and the value of \$H^2_{max}\$ needs to be determined, because there can be overshoot in the frequency response, depending on the damping. If I have more time later on I might add details about second order systems. For the time being I hope that the answer is clear enough so that everybody can derive the noise bandwidth of any system they are interested in.