The dialing computer is a system created by the Tau'ri to allow dialing Earth's Stargate in the absence of its DHD, which was deliberately left behind by Dr. Ernest Littlefield when the Alpha Gate was transported from Germany to the United States. (RPG: "Fantastic Frontiers: Stargate Season One")
History
According to Samantha Carter, it took 15 years and three supercomputers to MacGyver the system. (SG1: "Children of the Gods") Despite this, the computer was initially very primitive: it allowed to dial only a single planet — Abydos — and travelers had unpleasant experiences when exiting the Stargate on the other side. (Stargate)
Throughout years, the computer has been upgraded several times. The first improvement was a database of Stargate addresses calculated from the Abydos cartouche found by Daniel Jackson, programmed by Carter to update addresses to account for stellar drift. (SG1: "Children of the Gods") This database was later complemented with new addresses extracted by Jack O'Neill from the Ancients' repository of knowledge, which was highly prized because most of them were unknown to the Goa'uld; O'Neill also upgraded the dialing program to allow utilizing the eighth chevron, enabling intergalactic travel provided enough power. (SG1: "The Fifth Race")
Soon, Carter sped up the dialing to secure the Stargate from an off-world activation by Sokar. (SG1: "Serpent's Song") However, as the dialing computer continued (and still continues) to rely on manually rotating the ring and locking the chevrons, the system by design never approached the speed that could be achieved by dialing with a DHD.
After the Reetou invasion of the SGC, the dialing computer was equipped with fingerprint scanners to prevent unauthorized personnel from using it. (SG1: "Show and Tell")
Before the year 2007 the Tau'ri had developed a portable version of the dialing program that could be installed on laptop and when used with a naquadah generator could dial any off world gate, that lacked a DHD. Presumably this version of the dialing program did not manually rotate the gate, but worked in a manner similar to an auto-dialer. (SG1: "Bad Guys") The third Alpha Site appeared to use a combination of this program and a DHD. (SG1: "Gemini") It is also the same method use at the Icarus Base but with slight modification to block incoming wormhole and to draw energy from the planet core as a power source. (SGU: "Air, Part 1") It is also assumed that currently all Tau'ri offworld base utilise this method as it can be seen in the Gamma Site. (SG1: "The Scourge")
Despite its drawbacks, the dialing computer continues to be employed by the SGC, due to the extra security and flexibility it provides over a DHD, and its compatibility with other SGC technology.