From what I understand, an oscillating reaction is a reaction in which the different concentrations of reactants oscillate in time. In the case of the Belousov-Zhabotinsky reaction, that amounts to seeing the solution change colors from clear to yellow and back every 4 seconds or so. The reaction itself involves 40 elementary reactions which makes it quite complicated!

There is another reaction of the same name that goes from red to blue every minute or so also. This uses something called ferroin (red) and ferriin (blue) to make the colors. I don't know exactly what you put in to make the reaction, but the basic gist of the reaction goes something like this: first, bromous acid concentrations increase exponentially in an autocatalytic reaction (it catalyzes itself). This reaction changes ferroin to ferriin (so we go from red to blue). When there is too much ferriin, it reacts with other stuff in the solution which causes it to turn back to ferroin slowly, releasing bromide ions in the process. Bromide inhibits the autocatalytic reaction of bromous acid so you get no new ferriin. In time, most of the ferriin becomes ferrroin (you go from blue back to red) and the bromide ions get captured again in. So we've gone from blue back to red, AND ended up at the starting point again. So the whole process happens again in a loop.

Like I said, the actual process is very complicated but these are the basics. By the way, if you do a search on the web you can find quite a bit of information about this reaction.

The Belousov-Zhabotsinsky oscillating reaction was discovered in the 1950s. Interestingly, it was dismissed initially because the reaction did not appear to go to equilibrium. It wasn't until years later (early 1970s)that the reaction mechanisms had been worked out. The details of the BZ reaction is somewhat intricate, and I will not try to describe all the details here (but I'd encourage you to look into it yourself if you're interested!).

There are basically three processes, or groups of reactions that drive the BZ reaction:

Process A: consumes bromide ions
Process B: Cerium is oxidized
Process C: Cerium is reduced and bromide ions are created.

Process B is an autocatalytic process that oxidizes Cerium. Autocatalysis means that one of the products of the reaction is also a catalyst for their action. In the BZ reaction, this autocatalytic process is comprised of a series of two reactions. Here, a product of the first reaction drives the second reaction, and then a product from the second reaction drives the first reaction. This leads to a "circular process" and the net result is a rapid formation of one of the products.

The key to this process is the concentration of the bromide ions. When the bromide concentration is high, process A is the dominant process. When the bromide concentration is reduced past a "critical point," the autocatalytic process in B becomes dominant and the Ce becomes oxidized. At some point, process B will become limited and slow down, such that process C kicks in. At this point, the Ce is reduced, and bromide ions are generated. When the bromide concentration reaches a certain point, process A will start up again and whole thing can begin again.

In this reaction, ferroin is usually present as an indicator of the oxidation state. As the oxidation state of Ce is increased, the oxidation state of iron in the ferroin is also increased and changes the color of the solution. Thus as the processes go from A to B to C, and then back to A, the color of the solution will oscillate. While the color of the solution is determined primarily by the oxidation state of the Fe, there is probably also a contribution from the Ce ions as well (although I'm not positive about this).

The simple answer to this is that while most chemical reactions have a simple path to equilibrium, it is possible to have reaction kinetics which actually favor the reverse reaction. For example, the energy profile can be very sensitive to the local concentration of some species that mediates one direction of the reaction --but which is subsequently transformed into another species mediating the reverse reaction.

There is a very nice article in Scientific American describing the action you mention, about 1975 or so. As I recall, it made the cover. Here are some current sites:
Belousov-Zhabotinsky reaction (0ver view)
BZ reaction
This is a fun reaction to do yourself -- you might ask your chem teacher to get the chemicals -- the recipe is very simple...

An oscillating reaction is a chemical reaction that appears to alternate between two compounds or products. Stirred solutions might be seen to change color in a periodic, or oscillatory fashion. Unstirred solutions might show regular wave-like patterns of color which propagate through the solution. These effects originate from local variations of chemical concentration within a solution.