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? 
If you want to avoid algebraic geometry and topology, then you will probably be forced to use algebra.  This is okay, because the things you ask about have algebraic interpretations.

First of all, it probably helps to write the function field as $k(x,y),$ where $x$ and $y$ are related by some equation $f(x,y) = 0$.  One problem is going to be that $f$ can't be chosen to be smooth in general (because not every curve can be embedded as a smooth plane curve), but it's probably best to ignore this; if it becomes important, then probably your colleagues will be delving deeper into the theory anyway, and so the whole level of explanation can be ramped up.

Now to explain divisors, you can have them imagine intersecting $f(x,y) = 0$ with some other curve $g(x,y) = 0$; the intersections will be a bunch of points, possibly with multiplicity.
This is a divisor.   So divisors are just natural ways of encoding how curves intersect one another.  (Again, I am ignoring here the issue of points at infinity; you will have to decide whether that is appropriate, or whether you need a higher level of precision in your explanations.)

As for the genus, it's a little more complicated to explain algebraically, but possible; here goes:

Suppose that the equation $f(x,y) = 0$ has degree $d$, so your function field corresponds to a degree $d$ curve in the plane.  Now let $V_n$ be the vector space of all polynomials of degree $\leq n$ in $x$ and $y$.  Let's suppose that $n \geq d$ (and in general, we should
think that $n$ is large).  

A simple computation shows that $V_n$ has dimension $(n+2)(n+1)/2$.  Inside $V_n$, we have
a subspace consisting of all the multiples of $f$.  This subspace is obtained by taking
the elements of $V_{n-d}$ (i.e. polynomials of degree at most $n-d$) and multiplying them by $f$, i.e. it is the subspace $f V_{n-d}$ of $V_n$, and so has dimension $(n-d+2)(n-d+1)/2$.  If we look at the quotient $V_n/fV_{n-d},$ this then has dimension 
$n d + 1 - (d-1)(d-2)/2$.  

What is the meaning of this quotient?

If $g \in V_n$ represents a non-zero element of $V_n/fV_{n-d}$, then it is a degree $n$
equation that is not divisible by $f$, so it doesn't vanish identically on $f(x,y) = 0$, so by Bezout's theorem, the intersection of $g(x,y) = 0$ and $f(x,y) = 0$ is $n d$ 
points, i.e. a divisor of degree $n d$.  So we see that non-zero elements of 
$V_n/f V_{n-d}$ correspond to those divisors of degree $n d$ that are obtained by 
intersecting $f(x,y) = 0$ with a degree $n$ curve.  (In fact, we should think about non-zero elements up to scaling, because if we multiply $g$ by a non-zero scalar,
the curve $g(x,y) = 0$ doesn't change.)

Thus among all the degree $n d$ divisors on $f(x,y) = 0$, those that come by intersecting with a degree $n$ curve form a space of dimension $n d - (d-1)(d-2)/2.$  (Here I have
subtracted 1, because rescaling accounts for 1 of the dimensions in the above formula.)

Now what is the dimension of the space of all divisors of degree $n d$ (with non-negative coefficients,
which are the only kind that can possibly arise as intersections; divisors with non-negative coefficients are called effective)?  Well, we just have to choose $n d$ points and add them together.  We are choosing the points from a curve, which is one-dimensional, so that means there is an $n d$-dimensional space of effective divisors of degree $n d$.

Thus we see that, in the $n d$-dimensional space of all effective divisors of degree $n d$,
those that arise by intersecting with a degree $n$ curve are a subspace of dimension
$n d -(d-1)(d-2)/2$.

The quantity $(d-1)(d-2)/2$ is precisely the genus.
So what we see is that the bigger the genus is, the harder it is for a degree $n d$
divisor on $f(x,y) = 0$ to be obtained by intersecting with another curve.

For example, if the degree $d$ is $1$ or $2$, then the genus vanishes, and 
every degree $n d$ divisor comes from intersecting with a degree $n$ curve.
E.g. on a conic (i.e. when $d = 2$) any two points come from intersecting with a line
(the line that passes through those two points), any four points come from intersecting 
with a conic, and so on.

On the other hand, if the degree $d = 3$, then not all triples of points on $f(x,y) = 0$
come from intersecting with a line: in fact if you give yourself two points, then
they determine a line, which in turn determines the 3rd point of intersection.
Similarly (going up to the case $n =3$), a general set of 9 points on the cubic doesn't come from
intersecting with another cubic; instead, if you give yourself 8 points on the cubic
curve, you can find another cubic passing through these 8 points, and its 9th point of intersection is uniquely determined by the given 8 points.  (Since 9 = 8 + 1, this 
is a particular manifestation of the fact that our cubic curve has genus 1.)

As you can see, I've lapsed into a more geometrical way of thinking (using notions such as dimension), but I don't think one can avoid this completely: the concept of genus arose historically from essentially the kind of computations that I've just been making, and at some point you have to think about spaces of divisors and their dimensions if you want to understand it.  Still, I hope that this gives you an avenue to explaining the concept which is more algebraic, and so more accessible to your colleagues.

One more technical remark: The space $V_{n}/f V_{n-d}$ is an example of a Riemann--Roch space, and the formula for its dimension ($n d + 1 - $ the genus) is a special case of the Riemann-Roch formula.

[Technical remark added in response to a question of T. in the comments below; feel free to ignore it if it is at too high a level:]  Note that the above discussion works even if $f$ is allowed to be singular (and irreducible, say, so that
we can't have polynomials $g$ which vanish on one component of $f(x,y) = 0$ without vanishing 
on the whole curve).  The reason is that $(d-1)(d-2)/2$ is always the arithmetic genus of a plane curve of degree $d$, whether or not the curve is smooth, and it is the arithmetic genus that intervenes in the Riemann--Roch formula.  (I believe that this is the origin of the adjective arithmetic in arithmetic genus: it is this version of the genus for singular curves that comes up when you make Riemann--Roch-type calculations of the dimensions of various spaces of divisors.)
