It usually represents a rank. i.e. White with a stripe (either a new belt with a stripe in the middle, or sometimes just black tape at the ends) it a higher rank than just a white belt. Yellow is higher than white w/ stripe, but yellow w/ stripe is higher than yellow. In some styles that use tape stripes rather that a new belt with new colors, there may be multiple stripes before a new belt is given.
The one exception to this is that in some styles, on the color before black (usually brown or red) there may be a stripe that stands for a title, rather than a rank (though the title still makes gives the person higher authority over others who don't have the title). In the style I trained in, for example, the brown belt had 4 stripes. The fourth represented the title "kari-shodan," which loosely translates to probationary 1st degree black belt.
For the style I trained in, the ranking though black belt went as follows:
White (10th "kyu")
White w/ stripe (9th "kyu")
Yellow (8th "kyu")
Yellow w/ stripe (7th)
Green (6th)
Green w/ stripe (5th)
Blue (4th)
Brown w/ 1 stripe (3rd)
Brown w/ 2 stripe (2nd)
Brown w/ 3 stripe (1st)
Brown w/ 4 stripe (kari-shodan)
Black Belt
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