The greater the zoom range, the more difficult it will be to hold the
camera steady at the 'longest' setting ('zoomed right in').  It's also
likely that the lens will let through less light at the longest settings,
which makes the problem of 'camera shake' even worse.  Some sort of 'image
stabilisation' will be required unless you are prepared to use a solid
tripod.

Image quality is likely to be sacrificed in order to achieve an extreme
zoom range; whether that is enough to matter to you, only you can know.  

If there are particular models you have in mind, see if they have been
reviewed by magazines or on-line web sites - for example,
<http://www.dpreview.com/> and <http://www.steves-digicams.com/>.

If you are serious about using very long focal lengths ('zooming in' from a
great distance), consider using a camera that allows you to swap lenses.  
The results are likely to be worth the extra cost and trouble.

Only real tradeoff besides price is physical size.  Definitely worth the
money.  If you want a shirt pocket camera though you won't get it with a
12x zoom.

Focusing becomes more difficult at higher zoom settings.  Therefore
image stabilization becomes a necessity.  Otherwise you'll have to use a
tripod.  I have a Kodak P850 which has 12x optical.  It's an awesome
camera for the price ($299 at Costco with docking station and carry bag).

Depends on what you want to use it for.  I bought a 10x camera last year and
I rarely use the camera at that zoom.  When I do use it its not for taking
photos but just for looking into the distance.
The trade off of course is that most higher zoom cameras are bigger, so as
others have said, if you want a pocketable camera its better to go for the
lower zoom.

Check this out....

Sony DSC-H1 12x optical, hand-held (cropped):
http://www.cox-internet.com/kenb2002/MoonShot.jpg