An air compressor isn't a tool on its own, but a power source for air-powered tools.
Considerations When Choosing an Air Compressor
Most air compressors have a metal storage tank attached to them that they fill up when turned on and not in use, reducing how many on/off cycles it has to go through, especially when the connected air tool is used in many frequent short bursts. The capacity of the tank will affect how long the compressor can go before on/off cycles, as well as how much air you have to use after you turn the compressor off completely.
An Auxiliary Tank Set-Up by 802 Garage
This is the set-up used by Aaron of 802 Garage to expand the total amount of compressed air available to him while working. The external tank was added to the tank already on the compressor so that both add up to the total amount of air available. To see a list of all of his recommended parts, check the video description.
Additional or upgrade tanks are available to improve capacity on some compressors. This will require a custom set-up and the know-how to attach the tank. The tank will also have to be rated for more PSI than the compressor is rated for, just in case other factors change what the compressor can do or what the tank is holding, such as a noteworthy change in temperature over time while the tank is at full capaciy.
Due to how they work, it's very difficult to get an air compressor that is quiet, but some are definitely louder than others. Some are equipped with mufflers or other noise dampening devices. Placement within a shop or garage can also affect how audible the sound is.
Power, Pressure, & CFM
While power isn't an exact metric of how much air is moved, it is a requirement for higher pressures and CFM. The pressure will affect the power or speed of attached tools, while CFM will affect how much air it can move over time. If an attached tool uses air at a rapid rate, a high CFM compressor will be able to keep up with it, while a compressor with a lower CFM that can't keep up would mean that the tool would slow down with use, until it is stopped and reserve pressure can be built back up.
If your compressor is just going to sit in one place and never move, its weight isn't a big factor. However, if you plan on moving it around, especially if you intend to drive it places, weight will be an important factor to consider.
Size will matter when it comes to transportation as well, but unlike with weight, it is an important factor if it isn't going to be moved anywhere, because it will take up space in your garage or shop bay.
The smallest possible air compressor would be tankless, but not having a tank to act as a buffer for changes in tool use/requirements won't work well for most automotive tools. A "pancake" style compressor tank is a popular style of compact compressor and tank combo.