Can someone explain the ABC conjecture to me?

Try Mazur's Questions about Number (1995).

One simple, surely fundamental, question has been recently asked (by Masser and Oesterle) as the distillation of some recent history of the subject, and of a good many ancient problems. This question is still unanswered, and goes under the name of the ABC-Conjecture. It has to do with the seemingly trite equation A + B + C = 0, but deals with this equation in a specially artful way.

In Serge Lang's Algebra, he says: "One of the most fruitful analogies in mathematics is that between the integers and the ring of polynomials over a field". He then proves the abc conjecture for polynomials, and for good measure he proves Fermat's Last Theorem for polynomials. In other words, Lang is saying that if something is true for the ring of polynomials, one ought to check if it is true for that rather important ring called the integers. But it turns out that the ring of integers can be rather more troublesome, which may be surprising. So I'd say the abc conjecture is important because its proof over polynomial rings tells you it ought to be true for integers, but like Fermat it is rather more elusive than it appears. if you have access to Lang, his writeup in Chapter IV.7 is really good.

If one wants to avoid epsilons and constants in the formulation of the conjecture one can use this one instead.

If

i) $\mathrm{rad}\,(n)$ is the product of the distinct primes in $n$,

ii) $A,B,C$ are three positive coprime integers,

iii) $A+B=C\ $,

iv) $\kappa >1$,

then, with finitely many exceptions we have $$C<\mathrm{rad}\,(ABC)^{\kappa }.\tag{1}$$

For example at most finitely many instances of $C>\mathrm{rad}\,(ABC)^{1.005}$ are expected.

Addapted from The ABC-conjecture, Frits Beukers, ABC-day, Leiden, 9 September 2005.

On the other hand if one needs to find an implied or an effective constant, then the following formulation is better

For every $\varepsilon >0$ there exists $C(\varepsilon )$ such that

$$\max\left( \left\vert a\right\vert ,\left\vert b\right\vert ,\left\vert c\right\vert \right) \leq C(\varepsilon )\left( \displaystyle \prod\limits_{p\mid abc}p\right) ^{1+\varepsilon }\tag{2}$$

for all coprimes integers $a,b,c$ with $a+b+c=0$.

(From Enumerating ABC triples, Willem Jan Palenstijn, Universiteit Leiden, Universiteit Antwerpen, 26 November 2010)

Added. Here and here you can read two historical notes by Oesterlé and Masser.