What does the frequency band mean when it comes to finding aliases? The time signal which i'm trying to find the aliases for is: 

$$x:{\mathbb R}\rightarrow {\mathbb R}\\\   x(t)=\cos(50t) +2\cos(70t).$$

If the sample period is $T_s = \frac{\pi}{60}$ then according to Nyquist -Shannon sampling theorem (which btw my professor failed to prove) there is/are a signal(s) which after sampling will be equal to the sampled version of the above signal, if we sample those with the same sample frequency in the frequency band $[-55, 55]$.

I don't understand the meaning of the last sentence , what does a frequency band means here?
Let's use a simple example: A sine wave of 400 Hz. The Fourier spectrum has two components: one at 400 Hz and one at -400 Hz (since it's a real valued time signal).

Now let's sample this at 1 kHz. Sampling in one domain is periodic repetition in the other domain. So the sampled signal consists of your two original frequencies, -400 Hz and +400 Hz repeated again and again in 1 kHz intervals. So you get components at +-400 Hz, +-600 Hz, +-1400 Hz, +-1600 Hz, etc.

Any of these mirror frequencies (600, 1400, 1600, 2400, 2600, ...) will give you the exact same samples as the original 400 Hz.