ok, a word of explication on short notation in chess games.
Most of occasional and hobby players know the long notation, which is easy to read and very clear...
It consists on 2 positions separated by '-' which describes the movement of a piece.
Ex : e2-e4 : which means the pawn on e2 goes to e4.
There also a couple symbols used which you have to know :
K = King
Q = Queen
R = Rook
B = Bishop
N = kNight
x = capture (This symbol is often omitted when recording pawn capture moves). T
+ = check
++ = doublecheck
# = checkmate
O-O = castles short on the King's side (Often shortened to OO.)
O-O-O = castles long on the Queen's side (Often shortened to OOO.)
So if the Knight moves from b8 to c6 we would indicate this as 'Nb8-c6'
The difference with short notice is that we only indicate the 'arriving position' of the piece.
Going back to the previous moves
White would say : e4 , meaning that the pawn moves to e4, and since we are at the start of the game it's obvious that he meant that it is the pawn on e2 because it's the only piece that can move to e4
Black could answer : Nc6 , obvious also that he meant the Knight on b8, because he indicated that he moves a kNight and there is only one knight that can move to c6.
A word of caution about short notation :
There can be a situation where short notation isn't explicit: Imagine following situation :
Black has a pawn on e5
White has a pawn on d4 and f4
it's white's turn and he want to capture the black pawn on e5 -> if white enters e5 or xe5, we have a problem because we don't know which pawn white used to capture, in that case you have to be explicit with long notation : d4 x e5 or f4 x e5 !
The same situation can occur if you have 2 Knights attacking the same piece !
ok Daniel, let's play 
I retain your first movement : e2 - e4
My answer : e7 - e5
And lets go from there, because g3 could be correct and in that case it would be your pawn g2-g3. However, based on experience with openings I assumed that you meant Ng1 - f3. But it's not to me to assume what you mean so lets take back that move.
So we have each moved a pawn (yours on e4 and mine on e5 and let's continue from that.
And if you prefer we can use long notation if that is easier for you.
k
