Intuitive/geometric way of thinking about effective divisors? 
What is the motivation/intuition/geometric way of thinking about an effective divisor? 

I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for effective divisors. We can extend this in the obvious manner to get a partial ordering of divisors.
I would consider any explanation that introduces the concepts of Weil and/or Cartier divisors to be obfuscating the underlying intuition of the definition in more formalism...
 A: An effective divisor is just a formal linear combination of codimension one subvarieties of some scheme. The best way to understand that statement is with an example. Consider a quartic surface
$$
X = \textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^4 + y^4 + z^4 + w^4)}\right)
$$
Notice if we take any homogeneous polynomial in the quotient ring defining $X$, we get a codimension one subscheme. Notice I stated subscheme and not subvariety here, this is intentional. If we consider the divisor $D$ of $X$ as a subscheme given by
$$
\textbf{Proj}\left( \frac{\mathbb{C}[x,y,z,w]}{(x^4 + y^4 + z^4 + w^4, x^4(2x^2 + 3x)(x^2 + y^2)^2)}\right)
$$
then we can write it as
$$
D = 4[x] + [2x^2 + 3x] + 2[x^2 + y^2]
$$
So we've kept the topological information inside the brackets and moved the scheme theoretic information outside of the brackets and turned it into the coefficients of our divisor. If you really wanted, you could write down $X_i$'s as proj's of these graded rings, but I leave this as an exercise for the reader.
