What does the first $1$ mean in $\ce{Li}_2 1 ^1\Sigma_g^+: (1s\sigma_g)^2(1s \sigma_u)^2(2s \sigma_g)^2$?
The conventional labelling of molecular states starts at X for the lowest state and A, B, C for higher ones and a, b, c, for equivalent states of the same symmetry but different multiplicity; see Herzberg spectroscopy books for many examples. These labels, however, reflect the order in which the states were discovered and so do not reflect their relative energies. It is becoming more common to precede a molecular state's symmetry label by an integer e.g. $1~^1\Sigma _g^+,~ 2~^1\Sigma _g^+,~3~^1\Sigma _g^+$ to reflect the energy ordering of states with the same symmetry. This notation reflects the way atomic levels of the same symmetry are labelled as $1$s, $2$s, $3$s, etc.  

As an example, the conventional notation for the first four electronic states of  $\ce{C2}$ in order of energy is

X $^3\Pi_u$ ;  a $^1\Sigma_g^+$ ; b $^1\Pi_u$ ; A $^3\Pi_g$

which becomes 

$1~ ^3\Pi_u$ ;  $2 ~^1\Sigma_g^+$ ; $1 ^1\Pi_u$ ; $3~ ^3\Pi_g$

In atoms the notation for a set of transitions starting from the 3p level in sodium is  Na $3~ ^2P_{3/2}$ .