The most popular pumps for mid-range espresso machine are undoubtably those manufactured by the Italian ULKA company. The pump I rebuilt here is the ULKA H58, which is discontinued. However, many machines still contain this pump, it can be had as a replacement part from some suppliers (Cerini Coffee), and it's one of the simplest vibratory pumps to work with. Internally, it's much the same as the ULKA E8 series, though it's rated at 15 bar.
Rebuilds of some other pumps:
ULKA EP4 (French, but useful images and animations)
Step 1: Remove Pump Body From Solenoid.
The pump body is simply screwed into the solenoid at the head nut. It should come out easily enough with a wrench on the head nut and the solenoid held in your hand.
Step 2: Remove the Pump Head.
The pump head and body are held together by nothing but friction. Don't worry: it's supposed to be that way. Twist and pull at it until it comes free of the pump body. You should be able to pluck the shuttle from the innards of the pump now as well.
Step 3: Disassemble the Inlet.
It really is an ingenious design. Once the pump body is removed from the solenoid, nothing really needs unscrewing or unbolting. The simplicity of design is one of the reasons why these pumps rarely need to be replaced: just take them apart, clean up the parts, and slap them together again.
Anyway, pullout the inlet fitting. You should be left with the pump body, the shuttle return spring wedged inside it. Use a blunt implement like a chopstick or bamboo skewer (to avoid scratching the inside of the pump body) to carefully push the spring out the inlet end of the body.
Step 4: Disassemble the Pump Head.
Now it's time to take apart the pump head. This is where all the action is, really: it converts the reciprocating motion of the shuttle to a one-way pulsatile flow. (In electronics terms, it's contains the diode: it's a half-wave rectifier. Of course, they used to refer to diodes as "valves" for a reason...)
First, push the head nut along the outlet to reveal the pump head itself. The shuttle bumper will probably fall right off.
Step 5: Continue Disassembling the Pump Head.
The important part of the pump is also the most delicate, so be careful! The check valve is a tiny little snip of a thing, and it'll make you wonder how in the blue blazes it manages 15 bar pressures.
Carefully pick out the check valve gasket from the pump head and set it aside. Then, even more carefully shake out/pick out with a toothpick the check valve itself: it's a tiny rubber pellet on the end of a spring that looks all-too-easy to irreparably damage.
Step 6: Line Up the Parts for a Spiffy "exploded View" Photo.
You wouldn't take things apart and be left with no evidence of your prowess, now, would you?
Well, you don't necessarily take the picture yet. First you fix whatever turned out to be the problem, and your motivation for taking the thing apart in the first place. In most cases, it's simply scale and corrosion, which can be solved with a combination of 50:50 vinegar:water, cotton swabs, and steel wool. Just take care to fully rinse all parts before reassembly.
Oh, and take the picture, too!
Step 7: Reassemble and Test.
Hopefully your espresso machine is constructed such that a non-installed test setup like this is possible. It'd be highly annoying to completely install the thing and find out it's not yet working. Just be sure not to run it for too long at a stretch on a dry boiler: you did drain the boiler before you started monkeying around with this, didn't you?
Step 8: Optionally Adjust Output Pressure.
One thing you might want to do before you get the machine entirely together is adjust the pump output pressure. Somewhere along the water line between pump and boiler (and usually as shown quite near the pump) is an overpressure valve, which is essentially a check valve with an adjustable high cracking pressure. In an ideal case, this will crack somewhere around 10 bar, permitting brew pressures in the 8.5-9.5 bar range. However, in most pumps this is set too high.
First thing to do is mark the current setting with a Sharpie on both adjustment nut and casing. Then remove the valve core by loosening the adjustment nut fully, and swab out the inside with vinegar, flushing it well with water: a sticky overpressure valve can cause excessive and inconsistent brew pressures. Depending on the machine, the factory setting may be fine (especially if you've been getting good shots out of it). However, if the machine shipped with a pressurized portafilter, it's almost certainly several bar too high, and needs loosening. With most two- and three-turn OPVs, adjusting in 1/2-turn increments seems to be a good starting point. However, with no means to measure pressure, you'll have to assemble the machine each time and pull a test shot with known-good coffee.
See Home-Barista.com for more details.