It is no longer Intel vs AMD :). There are many more choices now, when you choose CPU for your computer. You could use Pentium4, AMD XP for single, AMD MP (or modified XP) and Xeon for duals. Xeons were expensive. So basically 2 choices for single: AMD XP or Pentium4. That was a past.
Now if you want 1CPU computer, you can have: AMD XP (2 types), AMD XP Mobile, AMD 64, Opteron, Sempron, Pentium4 (frontbus 533 or 800, cache 512KB or 1MB or 2MB), PentiumM, Xeon(different frontbus speed and different cache size) . That’s 8 different types of CPUs not counting the subtypes ! And all of them are good!

Intel Pentium4
This CPU (starting with 2.8 GHz) has special technology called HyperThreading, which makes the computer looks to Windows and Chess Software like Dual. So you cannot really install Fritz8 or Junior8 on computer like this. But HyperThreading can be disabled and it will act like standard single CPU computer. Cannot be used for Dual.

Intel Xeon
Server CPU from Intel. They can be used for Dual or Quad.

Athlon64
Athlon64 is a 64bit CPU. Good value, good performance, good for overclocking. Get the Athlon64 with 1MB cache (Athlons64 come with 512kB and 1MB cache).

AMD AthlonXP and AthlonMP
AthlonXP is a cheap and fast CPU. Cannot be used for Dual (you could use them in Dual, but hardware modification is required). AthlonXP Mobile are excellent too, especially the 2500+ and 2600+, very good for overclocking.
AthlonMP can be used for Dual without hardware modification, but it is more expensive than AthlonXP
AthlonMP is dead! Opteron1.8 retail (with fan and 3year warranty) is $250. This is price for 2xx Opteron, which may be used in duals. Dual Opteron1.8 will have 900kns with DeepShredder9.
My speed for dual AMD2600MP is around 1800kn/s for DF7 for big hash
(which I prefer – lately I use 256MB for DF7 and Sh704) and can be a little
above 2000kn/s when using 16-32MB hash.

AMD Opteron
A 64bit server processor from AMD. Can be used for Dual or Quad.
Opteron Quad currently (December2004) costs around $10,000.
What is Opteron 100, 200, 400, 800 family? Opteron is AMD competition for Intel Xeons. They are servers CPUs. Opterons 100 are for singles, 200 for duals, 800 for quads and 8cpu computers. Next processor in the family will get more expensive with the same processor speed. Basically they are the same processors, but that is the way AMD makes more money. There is no way to use 1xx Opterons series in duals (so far).
Opterons are difficult to overclock. If you want to overclock use Athlon64 instead.
AMD FX-51 is Opteron 148.
AMD FX-53 is Opteron 150.
Opteron 150 is cheaper than FX-53 ! (but FX-53 you can overclock)

Dual vs Single
Some 1cpu computers can get faster than 2cpu, they can also get more expensive then cheap dual. The fastest CPU can cost almost $1000 (both Intel or AMD), and a good CPU will cost $300-500. A cheap CPU for dual will cost $250 each ($500 for both). Xeons are not very expensive anymore. I really like the price for Xeon 2.8GHz with 800 frontbus. They cost only $235 (December2004).

Good 1CPU computer:
Use good CPU (Xeon, Pentium4 EE, or Athlon FX in 1cpu mobo) , overclock agressively.
Processors for single cpu computers from best to worst:
Opteron, Athlon64-FX, Xeon, Athlon64, Pentium4, AthlonXP, Septeron, Celeron.
The best deals now (February 2004):
Opteron 146 @ 2.0GHz, Opteron 148 @ 2.2GHz
Xeon 2.8GHz
Pentium4 3.2
AMD64 3400+, AMD64 3500+, AMD64 3700+
AMD-XP Mobile 2700+

Dont forget about memory
How much memory should I have?The more the better. If you have less than 1GB, go for 1GB. You have 1GB? Go for 2GB. Memory upgrade is always good if you have empty memory slots.