From one interested layperson to another, watch these excellent youtube videos. The jargon is at a minimum.
watch?v=7ImvlS8PLIo ('A Universe From Nothing' by Lawrence Krauss, AAI 2009)
watch?v=AHzbKrg2HIw (QED and richard feynman--three parts)
watch?v=ViQoUXu5uK0 (Quantum Physics for Dummies--two parts)
The gurus out here will give you a much more thorough and therefore complicated answer (or they might tell you that I've oversimplified or just gotten it completely wrong!), but I'll give you a basic layperson answer about what quantum means, and you should accept it skeptically. The basic idea is this question: is the universe continuous and smooth, or is it lumpy? Can space, matter, energy, and time be divided into indefinitely smaller units? The answer is no--the universe is lumpy, "quantized". Everything exists in discrete packets. Obviously matter is composed of discrete particles. But also space, energy, and time are also composed of discrete units. We discovered this about energy at about the same time that we were discovering it about matter. Energy comes in discrete packages called photons, which you probably already know, but also, a given photon can't have just any energy level. There are specific levels that can be taken, and the levels in between that you might think could be reached are simply unreachable. This is somehow related to the electron "shells" around an atom that you may recall from high school chemistry. If you hit an atom with a photon of the right energy, you can make one or more of its electrons jump to a higher shell. But if you hit that atom with slightly more of that energy, nothing will happen. Once you've made an electron jump up, then when the atom feels like it, it will allow the electron to settle, and in the process will release a photon with a specific energy relating to which shell the electron was in and which one it ended up in when it settled. So the basic idea is that energy exists in discrete packets, known as "quanta" (singular "quantum"). Hope this helps you, and I hope that in the simplification I haven't misrepresented the details. If I have, you'll know it soon enough because there will be a flurry of more informed (and probably cranky) responses.