How haste rating works
Haste is displayed in two different ways  as a haste rating, and as a percentage. Haste rating is what you'll find on your gear. Ultimately, the game sums your total haste rating, and then converts it to a percentage at a rate of 128.06 haste rating = 1% haste.
*Edited: As noted below in the comments, percentage haste effects such as Heroism/Bloodlust (+30%), Mind Quickening (+5%), Dark Intent (+3%), and Darkness (+3%) do stack multiplicatively. Contributions of haste rating from your gear stacks additively.
Many people misinterpret haste percentages by thinking them to be a discount, that is, that 1% haste means your cast of
Mind Blast will only take 99% the time it once did. This is not quite correct. You have to think of haste in terms of being a bonus, so that 1% haste means you'll be able to cast Mind Blast 1.01 times in the time it would normally take to cast 1.
How does this work in practice? Having 1% haste (128.06 haste rating) reduces the cast time of a 1.5second spell to 1.485 seconds; 10% haste (1,280.6 haste rating) reduces the cast time to 1.364 seconds; and 50% haste (6,304 haste rating) reduces the cast time to 1 second exactly.
For the mathematically inclined, the following formula shows how haste ultimately affects cast times:
Haste and the GCD: The haste cap myth
Before we go any further, I want to clear up a common misconception. There is no haste
hard cap. There are no haste plateaus. More haste is always better.
Always.
That said, there is a point at which haste becomes significantly less valuable as a stat. See, each spell a shadow priest casts triggers the global cooldown (GCD), a brief period of time in which we cannot begin another spellcast. This is most noticeable when attempting an instantcast spell; even though the spell's effect is immediate, we have to wait until the GCD elapses before casting another spell.
The GCD has a base time of 1.5 seconds. It is affected by haste, but it can never go below 1 second, no matter how much haste you stack. The point at which the GCD hits 1 second is the
haste soft cap. Hitting the haste soft cap requires 50% haste (6,304 haste rating at level 85).
The haste soft cap is relevant in theory. In practice, however, it's largely not. After all, raiding shadow priests currently have only about 15% of baseline haste (raised to
~20 21.1% via
Mind Quickening and
Darkness). The only time that the soft cap is even potentially applicable is during
Heroism/
Bloodlust. And with current values of haste being what they are, the brief period of Heroism/Bloodlust will push shadow priests just over that 50% haste line. Soft cap effects, if any, will be highly muted.
Haste and DOTs: The haste plateau myth
Without benefit of haste, understanding our DOT spells is a simple affair.
Devouring Plague has a 24second duration.
Shadow Word: Pain has a 18second duration.
Vampiric Touch has a 15second duration. The frequency of each DOT is the same  they all tick for damage once every 3 seconds. Simple!
Of course, since
Cataclysm went live, these spells do benefit from haste  and in a notsointuitive way. Haste affects both the total length of a DOT
and the time between ticks. For low values of haste, this happens in a simple way; at 5% haste, for example, DP ticks once every 2.86 seconds over a time of 22.86 seconds, for a total of 8 ticks.
When we start seeing larger values of haste, things get a little bit weird. See, DOT durations are indeed variable, but otherwise fixed in a range of +/ 1.5 seconds of their base duration. So, at point at which the total duration of DP would otherwise get cut to less than 22.5 seconds (6.25% haste), the spell gains an extra tick, with its duration resetting up near the maximum possible duration (now ticking every 2.82 seconds over a time of ~25.4 seconds, for a total of 9 ticks). The extra tick means more damage per cast and a longer duration of the DOT.
That extra tick is the genesis of the great "haste plateau" myth. Certainly, that moment at which your DOT duration jumps from 22.5 to 25.3 is a net DPS increase. Why? Simply put, it's because you don't have to cast Devouring Plague as frequently as you used to. The less time you spend casting DP, the more time you have to cast other spells (and the more mana you save). It's a wonky mechanism, to be sure, but it definitely works in our favor.
Here's the problem with the "haste plateau" argument, though: Just because there's a
huge DPS jump at 6.67% haste doesn't mean that getting to 6.68% haste isn't
better. Each point of haste rating will still make your DOTs tick faster, and it will still make your spells get cast faster. To repeat: More haste is always better.
Always. There's no point in shooting for a specific target when you do even more DPS by exceeding it.
To refresh or not to refresh
EDIT: Thanks to Tyler Caraway's incorrect advice and confirmation (sabotage?!), this section was edited in its entirety on May 26, 2011. Thanks to Xaydie for the correction.
As you may have been able to infer above, DOT frequencies and total lengths vary in a way to prevent the generation of fractional ticks on a single spell cast. That's the reason why extra points of haste continue to remain valuable to us past these "extra tick" milestones. But what happens when you refresh that initial cast?
When you refresh a DOT that's already active on a target, you're essentially telling the existing DOT to end on the next tick, with a full, new duration of the DOT to be tacked on seamlessly after. So, if you've cast Vampiric Touch (5 ticks over 15 seconds with 0% haste), and then refresh that cast at the 13 second mark, you'll get your tick at 15 seconds, and also get a full complement of five ticks for the new cast through to the final tick at 30 seconds. If the haste rating changes between casts (through, say, the Hurricane enchant), the DOT will begin to tick faster or slower after the next tick.
These mechanics suggest that you're best off refreshing your DOT after the secondtolast tick has landed, but before the final tick does. Refresh your DOT too soon, and you've inserted more casting time into the encounter than was necessary, resulting in a net DPS loss. Refresh your DOT after the initial duration ends, and you've lost a fractional second that could have been counted towards your next tick. Do it once, and it's no big deal; do it on a regular basis, and you could lose thousands of DPS over the course of an encounter.
The magic numbers
So, as you can calculate, there are many, many values of haste with added significance:

6.3% haste DP gains an extra tick

8.3% haste SW:P gains an extra tick

10% haste VT gains an extra tick

15.4% haste Heroism/Bloodlust pushes us over the soft cap

18.8% haste DP gains a second extra tick

25% haste SW:P gains a second extra tick

30% haste VT gains a second extra tick

31.3% haste DP gains a third extra tick

41.7% haste SW:P gains a third extra tick

43.8% haste DP gains a fourth extra tick

50% haste Haste soft cap; VT gains a third extra tick
Because each of these points represents sharp increases in the value of haste, theorycrafting simulations (such as
SimCraft) may show that haste gets relatively less valuable immediately after surpassing them. In part, it seems as if this may be the case, but these points when other stats are valued higher than haste are fleeting. Stack a few more points of haste, and all of a sudden, you'll see haste on top again. It's just part of the frustratingly complex calculus that makes understanding shadow priest stats a nightmare.
In the long run, these magical haste points make for interesting theorycrafting discussion, but they shouldn't affect the way you gear or play the game. For now, more haste is almost always better  there's just not reason to actively stack something else.
Are you more interested in watching health bars go down than watching them bounce back up? We've got more for shadow priests, from Shadow Priest 101 to a list of every monster worth mind controlling and strategies for raiding Blackwing Descent.