Modern espresso machines utilise a pump to generate the 9 bar of pressure required for espresso extraction.  These can either be of the vibrating type, or in more expensive equipment they may be rotary pumps.

Commercial lever machines rely on a powerful spring to generate the require 9 bar of pressure (unlike home lever machines along the lines of the Pavoni, etc, which have no spring return mechanism and instead rely on the operators hand).  This page addresses only commercial spring return lever machines.

Safety note:  The spring in a lever machine is powerful - unfettered, it  has been known to kill, and can easily break your jaw.  Always ensure that the portafilter is securely locked into the grouphead, and be aware that with an incorrect grind or no coffee present in the portafilter, the lever can rise at an alarming rate - ensure that no part of your anatomy is within striking distance of the lever - preferably operate the lever at arms length.

Single boiler machines
One of the inherent problems with single boiler machines is that the temperature required for effective steam production is higher than that required for espresso production.  On a modern domestic pump machine this is usually addressed by utilising solutions that employ two thermostats, one for each required temperature.  This usually results in a wait between steaming and espresso production to allow the boiler to stabilise at the required temperature.  The Gaggia GX lever machines employed a large exposed chrome group that acts as an effective heat sink, drawing heat away from the water as it enters the grouphead.  This results in water that is 88-90 degrees, good for espresso, whilst maintaining the 1.2 bar pressure (and therefore higher boiler temperature) that is ideal for steaming.  This means that unless espresso production approaches the order of 300 per hour, the groupheads will be able to keep the brew water at or near the correct temperature.  Machines that don't employ a method of succesfully reducing the brew water temperature should be adjusted at the pressurestat so that the boiler is kept at 0.9-1.0 bar, which may result in slightly less steaming power available.  My Gaggia machines are adjusted so the boiler is kept at 1.1 - 1.2 bar.  Obviously, being older technology, we are unable to replicate the 0.1 degree temperature accuracy that some advocate, but I'm still surprised by the accuracy of my machine.

Lever vs Pump and some history
This page wasn't written to set out and prove whether one type of espresso machine is better than another.  There are many pros and cons to the various types of espresso machines available today.  Historically, the lever machine was the mainstay of commercial espresso bars for 20 years, since the '40s when Gaggia invented the lever mechanism to get round the problem of earlier steam driven machines (invented by Bezzera and manufactured by Pavoni) that, by definition, overheated the brew water.  Achille Gaggia's machine (based on Cremonesi's design) of using a spring to generate the pressure necessary for espresso extraction resulted in "caffe crema" in the true sense, as the presence of the dark reddish brown crema indicated a well extracted shot.  This method was employed until the modern pump machine was developed by Faema in the early 1960s, although lever machines continued in popular use into the 70's.  You would think that as all technology supposedly improves on the previous generation, there would be little point in discussing lever machines or bothering to use them.  In some respects this would be true, certainly temperature control is easier (although still the bane of domestic machines), and the cost of implementing and manufacturing the heavy brass and chrome groups found on my Gaggias virtually prohibits any likelyhood of a domestic lever machine being built to the same standards.  However, in all other respects the lever machine is cheaper and simpler in construction and design.  There are few moving parts, no electronics/electrics apart from the auto level sensor and the pressurestat switch.  These parts are cheap and easily exchangeable, so once the machine has been built, longevity of service life is virtually guaranteed.  Servicing consists of routine gasket and seal changes.

The $64,000 question is always "is a pump machine's output better than a lever machine?"  In truth, the answer should be yes, since things like temperature control and consistent pump output affect the quality of the espresso, and large boilers can suffer from the water staling.  I am biased, since I use these lever machines, but I personally believe that a shot well prepared on my machine will equal any from a pump machine, particularly in the domestic environment (not surprising), and the few commercial pump machines I've used haven't produced better espresso.  I don't doubt that at the level of the famed La Marzocco my machines would pale, but I'd rather look at my (much cheaper) lever machines with their acres of chrome, much like the Faema E61 group attracts it's band of admirers, although technology has marched on.  I do think Gaggia got it right with the heatsink properties of the exposed group, and I certainly measure consistent temperature of espresso output.  My 2 group machine has produced several hundred consistant shots in a single day.

Operation of the lever
My Gaggia machines firstly use boiler pressure to admit water into the chamber above the coffee cake, allowing for preinfusion.  This occurs when the lever is locked or held in the down position, at which time the spring is compressed.  During this phase water can be heard entering the chamber.  If the grind is right, a drop or two of coffee will appear at the spouts after 5-10 seconds.  When the lever is released, the grind will determine the rate at which the lever rises.  For a single espresso, the lever should rise in approx 20-25 seconds, depending on the amount of preinfusion  we allow to occur.   As far as I can tell, Lever machines vary as to whether they produce singles or doubles (the size of the chamber) - my machines admit 1oz when the lever is lowered, whereas others tell me that their machine admits 2ozs, permitting either a double or two singles to be made.  As I only make doubles, this entails two pulls of the lever, and a (slight) adjustment to the grind so that the lever rises in 12-14 seconds, given that the single and double baskets have different size exit holes in the bottom.  As the lever nears the top of it's travel I pull again, admitting the next 1oz of water and then release the lever again.  As the system is completely manual, I remove the cup as the stream lightens, thus ensuring no overextraction enters the cup.

So, with the grind setup correctly, the order of operation is:

Load portafilter into grouphead.
Pull lever down (some machines have a lock in this position)
Water is admitted (can be heard)
Wait until water has entered chamber
Wait 5-10 seconds - espresso will either appear at the spouts if the grind is perfect, or not :)
Release lever
Lever should rise over a 20 second period approximately (for a single on my machine, a double if your machine dispenses 2oz water per pull) - gauge the colour and thickness of the crema.
Remove cup as stream starts to lighten (lighter spot becoming visible on the forming crema in the cup)
Several small adjustments of the grind may be required to achieve the above.

If your machine dispenses 1oz water per pull, then for a double:

Adjust grind so that the lever rises in 12-14 seconds
Pull a second time, allowing a few seconds for the next 1oz of water to enter, then release.
(try to do this fluidly, without interrupting the flow too much)
Remove cup as stream starts to lighten.

I assume that it would be possible to admit virtually as much water as you wanted, by virtue of adjusting the grind so that water flowed through the puck at a quicker rate during preinfusion, allowing more water in, but I can't see how this would end in a shot that was within the espresso rule of 20-30 seconds.

If anyone out there is an Italian with 20 years' experience at the lever, please write to me with your views :)