How can the outermost electrons in silicon be 2s and 2p electrons? My book says the number of outermost electrons in silicon is $4$. Yes, right. But then it says those are $\ce{2s}$ and $\ce{2p}$ electrons. The electronic configuration of silicon is $\ce{1s^2  2s^2  2p^6  3s^2  3p^2}$.

How are the outermost electrons $\ce{2s}$ and $\ce{2p}$ electrons. Those belong to third shell.

Then it says the maximum number of electrons in the outermost shell is $8$. Correct. But then it says those  are $\ce{2s + 6p}$ electrons. How?
The book means to say that there are 2 electrons in the  $\mathrm{3s}$ subshell and and 2 electrons in the  $\mathrm{3p}$ subshell. You can see this in the electron configuration as 


  $\mathrm{...3s^2 3p^2}$


and that the maximum configuration has 2 electrons in the  $\mathrm{3s}$ subshell and 6 electrons in the  $\mathrm{3p}$ subshell which would be 


  $\mathrm{...3s^2 3p^6}$