Cutoff frequency: exactly -3 db of power
The 3-dB cutoff is just one commonly used way to describe a filter. For some applications you might want to specify the 10-dB cutoff or 60-dB cutoff instead. It is convention that if someone says "cutoff frequency" without being more specific, they are talking about the 3-dB cutoff. You should think of the 3-dB cutoff as the frequency where the filter begins to roll off, not where it is strongly attenuating the signal.

There are a couple of practical reasons to use the 3-dB point as the conventional cut-off frequency.


In a one-pole RC filter, the 3-dB frequency is conveniently found at a radian frequency of \$\dfrac{1}{RC}\$.
If you draw a piecewise linear approximation to your one-pole transfer function plot, using a horizontal line segment for the low frequencies, and a 20-dB per decade sloped line for the frequencies well above cut-off, these two approximate lines will meet at the 3-dB frequency. Often when we just want to sketch a frequency response we make a Bode plot, which uses exactly this approximation. 




(image courtesy Wikimedia)