What exactly is flash sync speed and should it be a factor in a buying decision?
Flash sync speed is the maximum shutter speed possible when using a flash. For most flashes, the flash sync speed, sometimes also referred to X-Sync speed due to the use of Xenon in the flash bulb itself, is around 1/200th to 1/250th of a second. When using flash, your maximum shutter speed is limited to the flash sync speed. In many cases, this is perfectly adequate, as the flash pulse itself is sufficiently short enough (around 1/1000th of a second), and brightly lights up the scene beyond the normal ambient lighting for a fraction of the time the shutter is actually open. Flash sync speed can sometimes be a limiting factor, such as for action photography, as 1/200th or 1/250th of a second may not be enough to stop some kinds of action being photographed when fill flash is not able to overcome ambient lighting.
Some higher-end camera gear is capable of higher flash sync speeds. Some models support up to 1/500th of a second, which is better for photographing action. There are also alternative flash sync modes for better camera gear and flash gear. Normally, flash is synced with the "forward" shutter edge, and fires when the forward shutter curtain edge has opened and is moving. An alternative sync mode that syncs flash with the "back" shutter edge. When shooting with back curtain flash, you can produce action ghosting, and freeze your subject at the very end of the exposure, which is sometimes a desirable effect for sports photography. Finally, there is "high speed sync." With this alternative sync mode, cameras may sync to flash at any shutter speed, even up to 1/8000th of a second on top-end models.
High speed sync does have some limitations. With normal flash sync, there is a single pulse of the flash. In high speed sync mode, the flash pulses continuously thousands of times a second. This ensures that the scene is illuminated for the duration that the shutter is open and accommodates the behavior of a camera shutter at such speeds. The drawback here is that to provide enough power for continuous flash pulses, the power of each flash is less, by around 1 stop per stop of higher shutter speed. Additionally, since the scene is illuminated continuously for the duration the shutter is open, the flash itself is not as useful for "stopping" action. This is often not that big of an issue, however, as the higher shutter speed itself is capable of freezing action (particularly at 1/4000th or higher.)
If you don't need high-speed flash sync, any flash supporting a standard sync speed will suffice. If you need to sync flash at extremely high shutter speeds, then you will need both a camera body and a flash that supports high speed sync. You won't be quite as limited with high speed sync, but keep in mind that the power of your flash will be a little less than normal. (Generally, this is not a problem at all, and you can usually open your aperture to compensate...but it is a factor to be aware of.)
Short Answer: Probably not a Factor in Buying Decision
The sync speed is the fastest shutter speed you can use when using the flash.
Most flashes will sync with most cameras at around 1/250.
freeze the action:
This does not typically affect your ability to freeze the action, because (as @che points out) the flash only lasts for 1/1000 of a second (typically), so for images where the subject is being illuminated mostly by the flash, it will be frozen as though you were using a shutter speed of 1/1000.
Confusing? I find it a bit confusing, to be honest, but it does all work out logically once you get the hang of it!
X-sync is the lowest shutter speed during which the shutter is entirely open at some time, and thus allows use of flash. (You don't want it to light just top half of frame.)
X-sync differences don't play that much role in stopping action (flash pulse duration is around 1/1000 anyway), but rather in eliminating ambient (or overpowering sun, if you want) with flash in photos when you have both kinds of light.
If you have a flash-lit portrait outside, you usually want to have the surrounding landscape a bit darker than the person. Now, if you use a full power of your flash, and get proper exposure of foreground at f/8 and ISO 100, out might even get background overexposed at 1/200 sec if it's sunny day. Being able to go to 1/500 might make the shot possible, or allow you to raise ISO to 200 and save some flash power to get faster recycle times.
Sometimes you can kind of "cheat" by using high speed sync which fires the flash multiple times to cover all parts of the frame. This doesn't really help in this situation as it eats flash power you need to have foreground exposed properly.
IMO, it can be a factor in a buying decision, but when/if it is, you'll generally know it ahead of time. That is to say, if you've been running into problems (e.g., when using flash under daylight) and wishing you could get a faster flash sync, then it can certainly be worthwhile to get a body that syncs at higher speed. On the other hand, if you haven't run into a problem, then chances are pretty good that you really don't care.
At one time, X-sync speed was a serious consideration. When most cameras only synced at up to 1/60th or 1/90th, there were quite a few situations where it caused of a problem. The obvious problem arose when you had decent (but not really bright) ambient light. If you wanted, say, 1/500th to stop action, but only had ambient light to support (say) 1/125th, you ended up with problems either way -- if you didn't use a flash, you only get 1/125th, and with it blur. If you did use flash, you could only use 1/60th, so you had to stop down quite a bit and use the flash at (close to) maximum power to overpower the ambient light enough to keep it from leading to "ghosts". Shooting at full power, however, lead to longer flash cycle times, so you were more likely to miss shots as the flash recycled.
Unless you collect relatively old cameras, however, you'll probably get an X-sync of at least 1/200th, which is enough to prevent problems under most conditions. Being able to go higher is sometimes handy, but not all that crucial.
A few additional bits and pieces, though it's probably more in the range of trivia than useful information for most people:
A higher flash sync speed is useful if you wish to shoot with flash at large appeture (because you want low DoF) but are shooting at a subject that has bright backlight. A typical instance where this might happen would be a wedding shot with the sun behind the subject. You want flash because you want to bring up the dynamic range, you want large appeture because you want low DoF, and you have bright light so you want a shutter speed faster than 1/250th.
What is flash sync speed?
The flash sync speed is a limitation that's based on the shutter mechanism of the camera. Generally speaking, flash bursts can be much shorter than the shutter speeds of the camera. And with focal plane shutters, the shutter speed is determined by the gap between the first and second curtains as they travel across the frame. At the sync speed, that gap is big enough to leave the entire sensor/frame of film uncovered during the exposure. When you go faster than that, the gap is smaller than the frame, and you'll get black bars where the curtains cover the frame (with dSLRs, it'll be at the top and/or bottom of the frame). A typical maximum sync speed for dSLRS is around 1/200s.
High speed sync
To get around the limitation of X-sync speed, some camera/flash combinations are capable of high-speed sync (called HSS or FP [focal plane] flash)--the camera and flash communicate so that the flash can send out multiple bursts timed to follow the travel of the gap across the sensor, so that the whole sensor sees the same amount of illumination from the flash. But this will reduce the power output of the flash by roughly two stops.
High-speed sync becomes useful in two basic situations. (See also: Neil van Niekerk's Tangents post on when to use HSS).
So whether or not sync speed affects a purchase decision depends on the following factors:
Do you shoot flash?
If not, it doesn't matter.
Do you plan to use flash with a fast shutter speed?
If you're never going to use flash for fast-action photography where you can't kill the ambient, or you don't want thin depth of field in very bright conditions with fill flash, then you can probably limit your shooting at or below your sync speed. Studio shooting, for example, typically doesn't use HSS.
Does the camera you're looking at have a particularly slow sync speed?
1/200s as a sync speed is one thing. 1/160 is another (especially with cheap manual radio triggers that add a delay). And 1/10s is another. For example, the Panasonic GX-7 has a very respectable sync speed of 1/320s. With the built-in flash. It's 1/250s with an external flash. And in silent mode, it's 1/10s--so flash is completely disabled with an external flash. These are limitations you would probably want to factor in when deciding whether or not to buy this camera body.
Can your camera body do high-speed sync?
Not all camera bodies can. So, if, for example, you have a Nikon D3x00 or a Dx500 body, then the sync speed takes on more importance, because those bodies don't do HSS/FP, so you'll never get above that x-sync speed with flash. The GX-7 I mentioned above can do HSS with a four-thirds HSS-capable flash.
Are the flash and triggers you're looking at HSS-capable in your system?
Not all flashes or triggers do HSS, either. If you're using off-camera flash, then the camera, flash, and triggers all need to be capable of communicating the sync signals for HSS or you'll be limited to your x-sync speed.