In principle, an espresso machine is a simple device; it is designed to heat water to between 90°C and 96°C, and then push it through a puck of ground coffee at a pressure of 8 to 10 bar. The way a particular machine handles heating the water and creating the requisite pressure defines its type.
Spring Levers: This is the oldest system, introduced in the 1940s. A cylinder and piston system is used to pressurize the water. In many home machines of this type, the pressure is applied directly by the operator. The drawback of this is that it is very difficult to smoothly and exactly apply the required 40 to 50 pounds of force on the lever. All commercial lever machines and more sophisticated home machines use an uncoiling spring to power the piston. The operator compresses the spring, which does not require the force to be applied precisely. The reason a single espresso uses about an ounce of liquid is that this was the practical maximum amount of water that could be manually pressurized by this method.
In general, spring lever machines cannot be adjusted to deliver a precise pressure. They start at around 9 bar and, as the spring uncoils, smoothly diminish to around 7 bar by the end of the shot. This does not seem to adversely affect shot quality, and can in some case reduce bitterness.
This system applies pressure very smoothly, without the vibrations introduced by the rotary or reciprocating action of motor pumps. This difference may affect shot quality in two ways. First, it slightly reduces the amount of crema compared to motorized shots, although one very occasionally gets wonderfully creamy shots. Second, the taste of the shot is purer and more transparent, with less bitterness and acridity than otherwise identical shots from motor pump machines. How much of this effect is due to the other properties of lever machines, and how much is due to the lack of vibrations is unknown. But the actual difference in taste is quite apparent.
Rotary Pumps: The great majority of commercial espresso machines use rotary pumps, which can generate enough flow at 9 bar to serve multiple groups simultaneously. They are easily and precisely adjustable for pressure, and the pressure does not vary with the flow rates found in these machines. While they are not vibration free, they are smoother and quieter than the smaller vibratory pumps found on home machines. So, although they are a vast overkill for home use, some espresso enthusiasts get rotary pump espresso machines for their better adjustability and reputedly cleaner taste.
Vibratory Pumps: The home espresso market has exploded because of the vibratory pump, a cheap and small device that can pump just enough water at 9 bar to make a double espresso. Since these work on a reciprocating principle, they introduce far more vibration than rotary pumps. Much of this can be damped out by good overpressure valves and flexible piping, and better home machines have these. But still, they may produce a slightly less transparent taste than the other kinds. On the upside, the vibrations may create slightly more crema.
I am qualifying statements about the taste differences between vibratory and rotary pump espresso. When vibe pumps are properly adjusted, the reputed differences are contested, and in any case subtle. Also they may be influenced not just by differences in vibration, but also the different speeds at which each type reaches full pressure at the start of the shot.
Unlike rotary pumps, vibratory pumps produce a pressure that is strongly inverse to the rate of flow. If there are no controls, one must make a 2 ounce in 20 to 25 seconds espresso to get the pressure inside the 8 to 10 bar range. Smaller, slower pouring shots will have far higher pressures; larger, faster pouring shots will have far lower pressures. Better home machines have overpressure valves to limit the maximum pressure to about 10 bar, so that single and ristretto (reduced) espressos can be made without exceeding the normal extraction pressure range. Long shots, like Swiss café crema, will brew at 4 to 6 bar, no matter how well the vibe pump is controlled.