It is indeed possible to build an airplane from a kit, and in fact a number of companies would be delighted to supply you with the kit and tools you need to do so. The resulting plane is generally considered to be a sport aircraft or sportsplane, reflecting the fact that it is used primarily for recreation, not for transport of passengers. Before you build an airplane from a kit, you may want to check on regional regulations about using such aircraft; some regions require a pilot’s license or another form of certification to ensure that flying stays safe as well as pleasurable.
Kit airplanes have been sold since the 1940s, and it is also possible to build an aircraft from scratch, fabricating parts as needed and ordering specialty supplies. The advantage to building a kit airplane is that the kit arrives with all of the necessary parts, and sometimes the tools as well, and the aircraft has been tested, ensuring that it is safe to fly.
It helps to have experience when you build an airplane from a kit, both with flying and with assembling machines in general, as it can take some finickiness to get the airplane completed and safe to fly. While most kits come with extensive instructions and all the needed supplies, for people who have never engaged in major construction projects, it can get a bit confusing. It is also important to have a clear, clean workspace such as a garage, to ensure that no parts are lost and to keep the components of the aircraft in good working order as they are installed.
Homebuilt aircraft are, as a general rule, classified as experimental, and therefore they are subject to different regulations than regular aircraft. Being aware of these regulations when you build an airplane from a kit can help to avoid an awkward interaction with the long arm (or wing) of the law. Kit aircraft are also subject to noise ordinances and other rules in regional municipal codes, and researching these is also a very good thing to do before you build an airplane from a kit.
Many areas have associations of kit aircraft enthusiasts who love to take people up in their aircraft and talk about them. These groups can be good resources of information about learning how to build, handle, and maintain kit aircraft, and they may also have tips on good manufacturers to turn to for kits. Try searching for “kit,” “homebuilt,” or “experimental” aircraft and your region for a listing of such groups.