What are the important  positions where a player must reach first?

Obviously one should checkmate their opponent before they get checkmated.  

In general (and depending on position) the player who has the most move options has the edge.  Of course having more material can be important too.

In chess everything depends on the current configuration/position.  Finding the correct plan is more or equally important as finding the best move.

Hope this helps.  If not perhaps you could be a little more specific. Good luck.