I was filling in the forms to apply for a Schengen visa and I do not understand how to fill in my Member state of first entry. What does it mean exactly?
The Schengen zone has (as of 3 March 2016) 26 member states. And despite Schengen being a single zone for visa and movement purposes, the responsibility for approving Schengen visas is factored out to each of the 26 members.
Member state of first entry ... What does it mean exactly?
A person arrives into the Schengen zone from outside of the zone, like China or Egypt and so on. And the first time you touch down (or otherwise cross a border into a member state), it is your first entry.
At the risk of being pedantic, I can add that you can determine this by examining your itinerary and identifying the first place you will arrive in that requires a Schengen visa.
Also note that the the member of your first entry may or may not be the member who should process your application. See Which country visa should i apply to? - Schengen