Thermite generally refers to a mixture of a finely powdered metal and metal oxide in various ratios. Depending on the components, these mixtures can burn smoothly and slowly or explode with a bang (well, almost). Getting the components of thermite is quite easy, and there are several ways to do this.
First of all, pick your metal. Metals that have been used include aluminium, magnesium, zinc, and even ferrocerium (think camping firesteel). The size of the particles can range from coarse shavings (2 mm long) to a powder 100 times finer than white flour. The former are much less reactive and therefore much safer, while the latter have much more power and can be more interesting. Magnesium is the most reactive of these metals, followed closely by ferrocerium, then followed by aluminium and finally by zinc.
Then, obtain the metal. Aluminium powder is easily available in many sizes from online sellers. Here is 1 pound of 30 micron (very fine) aluminium powder: http://alphachemicals.com/inc/sdetail/165. For a smaller amount, here is 1 ounce of 200 mesh aluminium powder: http://www.elementalscientific.net/store/scripts/prodView.asp?idproduct=1363. Course magnesium turnings are easily obtained: http://www.mcssl.com/store/gallium-source/magnesium-metal/044---magnesium-shavings-454-grams. Magnesium powder is a bit more difficult to obtain (due to shipping regulations, I believe) but still is easily sourced. Here is 40 mesh magnesium powder: http://www.elementalscientific.net/store/scripts/prodView.asp?idproduct=2173. Zinc powder is also sold around the interwebz: http://www.ebay.com/itm/Zinc-Powder-1-lb-325-mesh-Purity-99-10-/190353593264. Ferrocerium shavings are too dangerous to sell, so they can be easily made. Find a knife, file, or similar object and preferably a low-quality Chinese firesteel, such as the ones found on bargain magnesium firestarters. Good quality firesteels like Swedish firesteel are extremely flammable and light at a very low scraping speed. This is bad because 1) it contaminates your metal powder mixture with metal oxides or 2) worse, it ignites the entire pile of shavings you are making, leaving you nothing to show for your work. Be always prepared for spontaneous ignition, and do not try to put out the resulting flame with water. To get zinc powder, open a fresh alkaline battery and get the center grayish part out. Wash the paste with water, then filter it and allow it to dry. This is impure zinc powder. To get zinc filings at home, take a piece of zinc casing from a carbon-zinc battery and apply a file or sandpaper to it. Zinc filings will be produced by the action. Since zinc is a harder metal than magnesium, it will be more work to create a pile of metal which is much less reactive than magnesium anyway. So magnesium is probably your best bet for tiny homemade thermites.
To get magnesium, get a preferably medium-quality magnesium fire starter (I use Coleman’s brand) and begin filing or sanding down one end. Ultra lightweight magnesium filings will begin to collect on the tissue below. Periodically dump them into a small storage container. Once you have reached your desired amount (which is about 3 grams for most thermites), cap off the container to prevent dispersal and store in a dry area. An expensive fire starter may make a nicer thermite but I have not tried due to price concerns, while some YouTube videos have shown some of the bargain fire starters being unable to light up.
Then we need an oxidizer. Thermites all use some form of metal oxide to act as the oxidizer in the reaction, thus negating the need for atmospheric oxygen. Many metal oxides can be used, including iron(II,III) oxide, iron(III) oxide, copper oxide, manganese dioxide, zinc oxide, titanium dioxide, and even finely powdered sand (silicon dioxide). Exotic thermites using vanadium pentoxide (which is very toxic) as well as molybdenum trioxide also work, but they are very expensive and performed for the sole purpose of isolating the resulting metal in an ultrapure state. Other thermites, such as lead dioxide (very toxic) and silver oxide (astronomical price: http://www.ebay.com/itm/Lab-Chemical-Purified-Silver-Oxide-Ag2O-10-grams-/320976242642?pt=LH_DefaultDomain_0&hash=item4abbacc7d2), are extremely violent and powerful. The iron oxides are probably your best thermite oxidizers. They have a high reduction potential, meaning that they make a reliable and energetic thermite. They are also dirt cheap (literally, they are some of the most common minerals on earth) so obtaining them can be as simple as dragging a strong NdFeB magnet (wrapped in a tissue) through beach sand and collecting the magnetite particles. These must be thoroughly crushed and mixed between two metal plates for best use. If you do not have a source of magnetite around, you can buy some easily (http://alphachemicals.com/inc/sdetail/147), or you may want to use hematite, another form of iron oxide. Artificial hematite, which is more reactive than magnetite, can be easily made using salt, electricity, and any small iron object. Dissolve salt in water and attach two iron objects to the sides of the container. Run any voltage higher than 3 volts DC and lower than 24 volts DC through the iron objects, with the larger object preferably connected to the positive and the smaller one connected to ground. All kinds of green gunk will begin to collect at the positive object, some of it turning to brown after a while. The negative object will just bubble. Run this for as long as you like and then filter the slop through a coffee filter. When it is completely dried, it will be completely red. Put the crumbly chunks of red iron oxide in an aluminium foil sheet and heat them in a kitchen oven at 350 F for half an hour to dehydrate them, then pour them into a container. Red iron oxide can be easily purchased, too (http://alphachemicals.com/inc/sdetail/225)
To make copper oxide, replace the negative iron object with a copper wire and the positive copper object with a large number of copper wires tied together at the top or a piece of copper. Run the electricity through as long as you desire. A yellow-orange solid will form throughout the container, turning green at times. Some blue will also be noticed. Then filter all of the copper pieces and copper oxide paste. Let it dry and heat it in the oven at the maximum temperature until it is completely black through and through. If it does not get completely black, make an aluminium foil dish and carefully place the powder in it, then heat it on the stovetop. Pour the black powder into a container. Copper oxide can also be purchased (http://alphachemicals.com/inc/sdetail/175).
To get manganese dioxide in an impure form, rip open a fresh alkaline battery (look on YouTube for instructions). Keep as much of the black gunk as you can. The gray inside stuff is not used in this step. The other metal oxides are not practical for low-price high-energy thermites, but they can be purchased [website]. Pure manganese dioxide can be found at http://www.ebay.com/itm/MANGANESE-DIOXIDE-1-lb-Pound-Lab-Chemical-MnO2-Ceramic-Technical-Grade-Pigment-/190634439641?pt=LH_DefaultDomain_0&hash=item2c62b2b3d9. Other manganese oxides are also used in thermites, but they are less reactive (http://alphachemicals.com/inc/sdetail/211).
Mix the thermite by grinding, where you can get creative. I sometimes use the tail of a pen to grind together the finely powdered metal oxide with the magnesium shavings, thus getting an even mix. Remember, the more finely ground and thoroughly mixed the thermite is, the better, hotter, and brighter it will burn.
To light your thermite, pile your thermite on a brick or similar surface and find an ignition method. You need to use a flame source hotter than butane (cigarette lighter). It is recommended to use short (1.5 inch) lengths of narrow magnesium ribbon to light the thermite. If the thermite is composed of an unreactive oxide (titanium dioxide, zinc oxide), you can consider placing some potassium nitrate crystals (Spectracide stump remover @ Home Depot) around the base of the magnesium ribbon. Bend the ribbon at the base so it can stand upright, then pile the thermite mix around it. The magnesium ribbon can be ignited by a butane lighter and a quick evacuation of the immediate area can be performed. The thermite should ignite as soon as the magnesium burns down to base level. Another common ignition system is potassium permanganate / glycerin. Both of these chemicals can be purchased easily online. When they are mixed, spontaneous combustion occurs after a waiting period of a few seconds to a few minutes. The thermite should ignite after the glycerol begins burning. Firework sparklers can also be used for ignition. Another method is a magnesium fire starter. Make a big pile of coarse magnesium shavings on top of the thermite mix. Use the provided ferrocerium to set the shavings on fire, and quickly move back. It is recommended to use heat-resistant gloves when doing this process. Another novel method is to place the thermite in a dry brown leaf (if you live in a deciduous area) and place some magnesium filings alongside the thermite (and overlapping it) on top of the leaf. Char the leaf – magnesium junction with a high-power magnifying glass. The black leaf char will absorb the heat, igniting the magnesium filings which will ignite the thermite. Use eye, face, and hand protection and back quickly away once ignited.
Warning: Iron thermites are very nontoxic to the environment and to people but they burn very hot and produce ultraviolet light. Do not look directly into a large number of thermites. Copper and manganese thermites are a little more toxic but can be disposed of normally. Beware of residual embers which may flare up when the ash is disturbed. Large thermites can explode if sprayed with water in an attempt to extinguish them.
Have fun with these fiery little reactions!