Framebuilding is the art and science of constructing the bicycle frame: that collection of tubes to which you attach wheels, cranks, handlebars, and much more. Frames are constructed in a variety of ways, including, but not limited to brazing, welding, gluing, and lay-up.
Alloy - Strictly speaking, an alloy is a mixture of a metal and another element. There are aluminum, steel, and titanium alloys. Strangely though, and this is worth noting, within the bike industry there is a tendency to refer to aluminum parts as alloy. We're not sure why this is; perhaps it's a translation thing. Regardless why, take care to make sure that what you are about it buy is what you want to buy.
Brazing - Brazing is a method more of less restricted to steel frames. Yes, you can braze aluminum, like this master craftsman, but within the framebuilding world, brazing is really for steel. There are essentially two methods of brazing a frame together: fillet brazing and lug brazing. Both of these methods tend to be done with an oxy-acetylene torch as the heat source and require the use of some sort of flux paste to prevent contamination of the parent and filler materials.
Dropouts - In the US, we typically just refer to dropouts and specify front or rear. Delving into the international scene, one can see the terms: frame ends and fork ends also used. In short, dropouts are the spots where the wheels attach to the bike. They used to simply be slots, but with the advent of thru axles, things have been blown wide open with numerous sizes and variations on the theme. Thru axles started out for suspension forks, but the advantages in stiffness and wheel retention have resulted in thru axles appearing on rigid forks and frames. 15mm x 100mm, 20mm x 110mm, and 12mm x 142mm are probably the most common.
Restricting ourselves to the world of steel dropouts, there are variations that determine how you can braze or weld a dropout to the seatstay/chainstay. Tab dropouts are probably the most popular with builders due to the creativity and various seatstay/chainstay tube sizes that they allow for. Lug style and plug style can be easier to braze but you need to nail the diameter of the seatstays and chainstays at the attachment points.
Fillet brazing involves joining two pieces with a melted filler as a "glue." Bronze is the most common filler material (often referred to as "brass"). After fillet brazing is completed there is often quite a bit of clean-up to be done using files and emory cloth. The more skilled the builder, the less clean-up there is.
Gluing - When carbon fiber first made its appearance on the bike scene, it most often appeared in the form of carbon fiber tubes glued into aluminum lugs. Today, if lugs are used to join carbon tubes, they are typically carbon fiber as well. The lug & glue method has seen a resurgence with the popularity of DIY bamboo bikes.
Lay-up - Lay-up is one of the methods for constructing carbon fiber frames. Essentially, a builder lays sheets of carbon fiber over one another in a very specific manner and then bonds it all together with glue (and sometimes heat). The most cutting edge construction methods used today are using fewer lugs and form the frame as one contiguous piece using molds and bladders.
Lug brazing involves the use of a lug which is essentially a socket. The two tubes being joined slip into the lug and brazing material, bronze or silver, is melted and pulled between the lug and tube by heat and capillary action.
Welding - TIG (tungsten inert gas) welding can be used to join steel, aluminum, and titanium. For all practical purposes you cannot weld dissimilar metals to one another. Welding is different from brazing because the tubes, aka the parent materials, are melted, typically along with a filler, in the joining process. Some builders will first join the tubes using a fusion weld without any filler and then follow-up with another round with the filler. Done correctly, there is very little clean-up involved with TIG welding.