When people say desktop computer, they are usually referring to a PC; although they may in some cases be talking about a Macintosh computer. In any case, the term is used for computers which are designed to be set up in one location and to stay there, unlike a laptop where portability is the idea. In the days when microprocessors were anything but micro, any computer which could be comfortably accommodated on a standard office desk was known as a desktop computer (and for the time, these were considered small). Today, people use the term desktop computer to refer to a specific kind of CPU casing. However, desktop computers come in many shapes and sizes from form factor models where the CPU sits behind the screen of the computer to the traditional tower style of casing.
The term 'desktop', then, specifically refers to a case that is horizontally-oriented. The desktop usually has the display screen placed on top in order to save space on the desktop. Most of the modern desktop computers of today have separate keyboards and screens. A specialized form of desktop case is used for home theatre systems. These cases incorporate front-panel mounted controls for both audio and video.
An all-in-one computer is just another name for a desktop computer that combines the monitor into the same case as the CPU. Apple has manufactured several different examples of all-in-one computers like the Macintosh 128K of the mid-1980's and the ever popular iMac of the late 1990's and 200's.Some older 8-bit computers much like the Commodore PET 2001, and the Kaypro II fit into this category as well. The all-in-one PC is usually more portable than the other desktop PC's and there have been many that were manufactured with carrying handles that were built into the case. All that has to be done is to unplug it and transport it to a new location.
However, all in one desktop computers suffer from the same problem as do many laptops - the difficulty of hardware upgrades. For instance, with iMacs, the CPU and other important hardware is permanently attached to the motherboard to save space.
Desktop computers have a number of advantages over laptops. One is that it is far easier and cheaper to replace hardware when needed, since these parts are much more standardized, which is not the case with most laptop computers. A desktop computer also features more room for expansion; a desktop computer typically has several PCI card expansion slots, compared to perhaps two on a laptop. These computers are also far easier to upgrade and repair; assembly and disassembly are far easier procedures and remains much the same regardless of brand.
Another advantage of the desktop computer is that the power consumption is not as critical as that of a laptop because the desktop computer is powered from the wall socket. The desktop computer also provides more space for heat to escape. The two large microprocessor manufacturers that happen to be AMD and Intel develop special CPU's for laptops because these consume less power and lower heat but they had a lower performance level.