What is the unit of SNR?
SNR stands for Signal to Noise Ratio. It is a ratio and as such does not have any units, it describes the proportion of signal to undesired noise. There is no single correct measure of SNR, it differs depending on the application.

In the equation you have given, the SNR is broken down in the following way:

1) Calculate the power ratio

$$\frac{\sigma_s^2}{\sigma_n^2}$$

Your equation requires us to know the variance of the signal and the noise, dividing one by the other gives the power ratio. To answer one of your questions, this is the ratio of power, not amplitude, we could achieve the same result by taking a ratio of the root mean square of the signal and noise - another measure of power.

2) Express the ratio in decibels

$$10\log_{10}$$

This is simply a convenience, the SNR may be a very large number so expressing the value on a logarithmic scale can make the information more manageable. Converting a unit to decibels simply involves taking the log10 of the value and multiplying by 10.