What are two molecular states ?

[xylai]

In molecular orbital theory, there are two molecular states. (see below)

I don't know the differences between these two states?

[alxm]

The difference is one of them has an apostrophe.

Seriously though, to my knowledge there's no system of notation for atomic or molecular orbitals which uses the prime symbol for anything in particular. I'm guessing in the context they're denoting a particular atom's 1s orbital with a prime and another's without.

Writing \sigma_g1s doesn't quite make sense by itself either - sigma is an MO, 1s is an atomic orbital, and you need at least two of those to get an MO.

\sigma_g signifies the even (gerade) symmetry, it's the same as the bonding sigma orbital.