Introduction
This article is rated difficulty 3 only because the timing belt must be removed to get access. If you're only removing the water pump without replacing the timing belt, the only difference is that you don't have to remove the motor mount. The motor mount can stay in place while you work around it. Once the belt is removed it's very easy, just remove 3 bolts and the pump will come out.
The water pump removal and installation procedure is the same on both ALH and BEW engines so this article is linked to both timing belt writeups. Ignore any small differences in the pictures vs. your engine because the procedure is the same. The actual water pumps are different though.
Caution - do not use green prestone coolant, you should use VW G12/G12+ or compatible coolant, see 1000q: coolant flush for more details.
10mm socket
water pump for ALH engine 1998-2003 VW# 038 121 011 a
water pump for BEW engine 2004-2006 VW# 038 121 011 g
I suggest a metal impeller water pump. The plastic impeller water pumps sometimes separate from the shaft. If this happens the engine will overheat with no other symptoms because no coolant is circulating. Replacement requires timing belt removal.
Drain the coolant as needed and remove the timing belt. For non pumpe duse cars, model years 1998-2003, ALH engine, refer to 1000q: ALH timing belt part 1-removal. For pumpe duse cars with BEW engine and a mk4 body, model year 2004-2006, see 1000q: timing belt replacement BEW - removal. I suggest replacing most of the coolant since most will be drained anyways. Have 3 liters/1 gallon of coolant and an equal amount of distilled water handy.
Open the radiator drain valve at the bottom corner of the radiator. If your car does not have one, remove
the lower radiator hose.
Optional: Remove the lower (or both) oil filter housing coolant hoses. Twist the
hoses to break the seal first. This will make it much easier to pull them
off. Then apply compressed air to the coolant reservoir. This will
blow out a little bit of coolant in the oil cooler and engine.
Remove the (3x 10mm) bolts holding the water pump. Here is a picture
of the water pump removed to show the location of the bolts.
Here is the backside of a metal impeller water pump.
If it's stuck, use a block of wood to gently tap it out, alternating on the top and bottom
until the water pump is loose. Do not hit it hard or use a screwdriver to pry it out because this will damage the metal surfaces. Then just
wiggle/pull it out.
Some
coolant will come splashing out, so have paper towels and a catch pan below the pump.
If it's stuck, dieselgeek sells a water pump removal tool which sits in the 10mm
bolt holes and pushes the water pump out. This is pictured below. 
Wipe away any excess coolant and leaking grease oil. Make sure the old
o-ring is removed!
To install the new water pump, use some coolant to lubricate the o-ring and press the new pump in place. Make sure to lubricate the o-ring or else it could twist or get pinched and spring a coolant leak. Do not use liquid gasket maker.
Torque the 3 bolts to 11 ft lbs.
To install the timing belt on non pumpe duse TDI, ALH engine, model year 1999-2003, see 1000q: ALH timing belt part 1-removal and 1000q: ALH timing belt part 2 - installation
To install the timing belt on pumpe duse mk4 TDI, BEW engine, model year 2004-2006, refer to 1000q: BEW timing belt removal and 1000q: BEW timing belt installation.
Do you have more DIY tips for removing the timing belt on an ALH or BEW VW TDI engine? Post your comments in the myturbodiesel.com forums