The best way to do this is using a jQuery plugin.
The most customizable and flexible one is jScrollPane which works for both vertical and horizontal scrolling and allows you to easily style it using CSS by applying styles to elements like handle, rail, wrapper etc. Note that scrollbals do require styling as they are quite ugly out of the box.
If you want something prettier try perfect-scrollbar which nicely mimics the behaviours of scrollbars to look like those on Mac OS X (nice and thing, visible only when needed).
I've also used jQuery-slimScroll and it also did the job very well. Lightweight, easy to use, good looking plus easy to style, but not as customizeable as the first one.
It is theoreticaly also possible to just hide the scrollbar using CSS only, but this is hacky and very bad for user experience. I would definetly go with a jQuery plugin solution as they will deal with default scrollbars all by themselves.