TITLE: Intuitive explanations for the concepts of divisor and genus
QUESTION [44 upvotes]: When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any intuitive explanations for these concepts, preferably explanations that are less dependent on knowledge of algebraic-geometry/topology?

REPLY [16 votes]: If your CS friends are like me, they might still find the answers above a little overwhelming. So you could start as follows: 
First, show them the Wiki's divisor page (it always works!). Then explain to them that by the fundamental theorem of arithmetic, any divisor is just a bunch of prime numbers with multiplicities. 
Next, tell them that completely similar to $\mathbb Z$, the FTA works over $\mathbb C[x]$ (which you can think of as a line). Except now that each "divisor" (polynomial) can be thought of as a bunch of points (roots of the polynomial) on that line, with multiplicities. 
But why stop as a straight line? One can do the same thing for a (reasonably nice) curve on the plane, and a natural way to get a bunch of points is to intersect with another curve. By the way, a bunch of points "divide" the curve, further justifying the terminology! 
Back to $\mathbb C[x]$, you can point out that the total numbers of points (with mult.) is the degree of your polynomial $p(x)$, or the dimension of the vector space $\mathbb C[x]/(p(x))$. 
At this point, if they still follow you, show them Matt E's splendid answer (-: