In the original movie, there exist only 2 Stargates, one on Earth, and one on Abydos. In the series, they expanded the concept to have many Stargates in a network. As such, many of the explanations for this aren't in the movie, and are from miscellaneous episodes of SG-1. I'll try to cite sources. I'll also only be discussing the Milky Way gates, as those are the ones that SG-1 deals with, which is where the vast majority of our detailed knowledge of the Stargate system comes from.
The way that dialing for the Earth gate works is they dial the 6 digit address for the remote gate, and end with the Point of Origin symbol. This was an issue in the movie because there are 39 symbols (link) on the Stargate, and they were trying to dial at random. Assuming they even knew that only 7 were needed, that's still roughly 39^7 dialing combinations they'd have to try. As there are 9 chevrons (spots where you can lock in a symbol), this would not be initially clear.
The symbols are derived from constellations in the sky on Earth. Each of these correspond to points in 3D space, so to dial a gate, you need to know which 6 points are around your destination. Alternatively, you can acquire a phone book with a list of valid destination addresses, which is how Stargate Command operated. They retrieved a list of formerly valid addresses from Abydos (link), and started testing them to find which were still valid. This provided them with planets to explore on an extremely convenient weekly basis. After stumbling upon an Ancient Repository of Knowledge, their listing of valid addresses was increased by Jack O'Neill while he had Ancient knowledge (link).
The point of origin symbol was a large stumbling block (haha!) in the original movie as the coverstone that prevented the Earth Stargate from working before it was buried was missing the point of origin symbol. The SG-1 show explained this was further complicated by the lack of a Dial Home Device (DHD). Without a DHD, a way to control the dialing had to be designed and constructed. In addition, they needed a full 7-digit address before they could dial out successfully.
The way the point of origin works is that each Stargate has 38 symbols that are identical on all gates (ignoring the Abydos gate in the movie, which had symbols corresponding to the constellations on Abydos). The 39th symbol is unique to each gate, and is the point of origin. If you have a DHD to dial, you enter 6 digits on the DHD, then press the large red button. This instructs the gate to then lock in the point of origin and establish a wormhole. Without the DHD, SGC had to find the point of origin symbol, which Daniel Jackson did in the movie. This allowed them to have their computer system that interfaces with the Stargate dial the point of origin.
As others have mentioned, stars move in relation to each other, so Stargates need some way to adjust for this. For any Stargates that have a DHD, they are part of a subspace network that communicate information about stellar drift, and make the minor adjustments necessary for smooth travel (link). As the Earth Stargate lacks a DHD, the first trips through the Stargate had side effects. Namely the people emerged from the gate cold and dusted with frost. Eventually Samantha Carter was able to make adjustments in the Earth dialing computers for this, and future travel through the gate was normal.