So one of the end results of this project is to build what is essentially a water heater and cooler. What the hell are they going to do with it?

Amplify DNA, yeah. Fun. If you've ever seen DNA, it's not that thrilling. (I'll give you a hint, you can get the same effect by sneezing after sanding drywall)

The really cool ingredient here is having the trained microbiologist in the lab, and the lack of insane standards to follow. Don't get me wrong, those standards make sense in many cases, but when your back is against the wall, you're going to find a way to make what you have on hand, work for you.

My earlier discussion of PCR is the quick and easy method. Take your DNA sample, mix in some things, and also mix in your primers, the A,T,C,G's that DNA is made of. Then go through the heating and cooling processes of PCR. Bingo, DNA amplification.

So, let's look at the advanced technique. There exist "DNA Tools" that cut DNA in specific locations. Dozens, hundreds of different enzymes that only cut specific locations of DNA. This is the technique that allows gel electrophoresis to work:

Cut two samples (or more) up with the exact same set of cutting tools. Then use a gel electrophoresis rig to "pull" the samples using an electric tension from a sample site, across the gel. The different sizes/weights of DNA will settle across the gel in distinct bands. Compare the bands to each other (take a look at the second picture). If one matches, that's your original sample's identity.

This takes awhile, and takes some more "stuff". Instead, lets see what else we can do.

Take a known sample. We'll say a known sample of The Dreaded Gombu, lethal creeping crud of NoWhereistan. Do your special procedure to extract DNA (Bacteria, viruses, etc, all of slightly different methods for extraction), get it in your test tube, and start PCR'ing it. We want to amplify it. A lot.
Next, we're going to take our special cutting tools, and cut this DNA up. And then we're just going to leave it. For a minute.
Now we go to our patient sample. This guy is sick, but we're worried he has The Dreaded Gombu. If we give him the treatment and he doesn't have it, it'll kill him (a little drama). We need to be sure.
We take our patient sample, extract the DNA from it. Keep in mind, that we're going to get ALL of the DNA out of it. The sick guy, all the little critters in him (we have more foreign DNA in our body than you want to think), and our mysterious disease causing agent. We put that mishmash of DNA into a test tube, along with some of our known, sliced up, Dreaded Gombu. NOW, we PCR the hell out of it. The PCR reaction is only going to efficiently work with the Dreaded Gombu slices. If we cycle this thing a bunch of times (each time doubles the reacting DNA, right?), we should see the amount of DNA increasing in the test tube, if there is Dreaded Gombu present. If not, we're not going to see much changing.

If you get this technique down, you can achieve more than a qualitative analysis. You can actually do a quantitative analysis of the DNA (How similar is this to a known sample?).

The technique itself is amazingly simple, but it takes a skillful hand (and in some cases, is a bit of an art, to really make the technique sing). The skill of the technician is very important, more important than the tool itself. For this reason, not having the tool is a stupid stumbling block, when the lab has the skills to use it.