Is the second energy level of the oxygen atom full? Explain

No it isn't because the atomic number of oxygen is 8.

Electronic configuration-

2,6 (=8)

The total electrons that can be accommodated is 8 in the second shell. There are 2 empty spaces.

Hence it is not full.

(And the valency therefore is 2)

Extra info:

According to the electronic configuration rules, the first energy level will have 2, then 8, 18, 32 and so on, but the electrons in the outmost energy level can only be till 8.