For example if I was defining the following function
exprod[n_] := Expand[Product[x + i, {i, 1, n}]]
Then why is the underscore after the variable n, necessary in the function definition? Where does this style come from or is it specific only to the Mathematica programming language?
The underscore comes from pattern matching.
The x_ matches anything and this anything is bound to the name x in the body of the function.
l[x_ * y_] := l[x] + l[y];
Then in l[2*z] first the expression 2*z is matched against the
pattern x_ * y_ . Then x is bound to 2 and y is bound to z.
Then the expression l[x] + l[y] is evaluated, and the result becomes
l[2]+l[z].
Now say we want to define the value of l on e to be 1.
Do we write l[e] := 1 or l[e_] := 1?
One says that l to (literally) the variable e must be 1.
The other says that l to something gives 1.
http://reference.wolfram.com/mathematica/tutorial/Introduction-Patterns.html