The lint screen configuration depends a lot on the specific dryer, but generally there is a lint screen which can be removed. This is the thing that you clean each load of clothes that you dry. Most of the time, it is a flat screen that slides vertically out from a compartment that is below the door of the dryer. If you take a flashlight and look into that space under the lint screen, you'll likely see a big buildup of lint. You can keep this maintained with a long, thin, tapered brush avaliable at home stores. When I clean this part, I use a bigger brush and a shop-vac stepped down to a 5/8" OD hose that can fit down there and get everything out. Sometimes you can take a panel off inside the machine that allows for easy complete access to this area, but normally it is a matter of poking around until you can tell by visual inspection that there's no more lint down there.
The heating element also varies in form. Most dryers are electric, and are heated by coils that act as resistive elements, such that when current flows through them, they create heat as a byproduct. This heat is channeled to be blown into the dryer by a fan. This is the part of the dryer where the fire will start. You must make sure this is as clean as possible. Often times you will see dark brown or black lint. It might look like someone spilled a bit of Guinness on the inside of your dryer. But this is not roasted barley, this is burnt lint. This is evidence of a near-fire in your dryer. Don't be too alarmed - this is pretty common. Normally the lint gets really hot and scorches, but no fire happens. But this area must be kept clean to prevent fires. At the same time, the coils (or gas burners and ignition system in a gas dryer) are fragile and delicate. It is a balance between cleaning that area and not disturbing the functional integrity of the device.
There are wires all over the place inside the dryer. It isn't exactly a functional test bed of nails for printed circuit board test, but it isn't quite as simple as those little DIY radio kits that your kids can do. Every dryer in particular will have an electric motor that drives a belt that turns the drum of your dryer. You know, the thing that causes it to spin so that the clothes move all around and get more evenly exposed to the hot air, so that they dry evenly and efficiently. Without getting all Electrical Engineering on you, this is basically just a big tight coil of wire that generates an electric field when current flows through it (it isn't as resistive as the heating element). The current causes a big magnet in the middle to spin with the current, which drives the belt, which directly drives the turning of the drum. If this gets too covered with lint, the electrical properties of the coil could be modified and effect the operation of the motor. If it gets extremely coated, wire connection points could become shorted by conductive lint. This must be kept fairly lint-free. It generally won't cause a fire, but should be maintained. It's good that this doesn't need to be "toothbrush clean" because it is usually really hard to get at.
Finally, there are air intake slits all over the back of the dryer. Often times you'll find a bunch of lint and/or dust behind your dryer. The back of the dryer is important to the function of the dryer, for this is the clean air intake for dryer operations. This lint and/or dust tends to be sucked into the dryer via the vent slits. To clean this, simply apply your vacuum to every slit back and forth until you no longer see lint or dust there. You may use your little brush attachment if you want, or the crevice tool. But for me the 5/8 tube works great, too.
That's the quick rundown on cleaning the inside of the dryer. OK, looking back through the writeup, it isn't that quick, but given the complexity of the system we're talking about, it is as quick as can be expected. If you just take the time to make sure all parts of your dryer are clean, you will get it right. At the same time, if you don't feel like spending a few hours figuring out how to take your dryer apart, which parts need to be taken apart, and how exactly to safely clean all the necessary parts, you can always give me a buzz (860.324.5213), and I will do all the hard work for you!