Steve's explanation of the "double stop" is closer to my understanding (compared to Ted's first response "
Teddahlia disbud the set of buds below the first ones(double stop)
Even as a cut flower grower; I disbud "two down" meaning I take the two (or 3) side buds AND the next buds (more accurately described as laterals) down on every stem. This isn't double stopping (as I define it anyway). "Stopping" is taking out the first terminal bud made by the center stalk of the plant and "double stopping" is taking out the first bud on the side branches (laterals).
As I have understood it, the goal of "double stopping" is mostly for timing. A dedicated show grower may want their plant's first bloom to go to a show because it is often the largest. Don't allow the plant to waste it's energy on a meaningless bloom that isn't going to go to the show. Some show growers keep meticulous records on the # of days between the time they can spot a bud until it has a mature enough show ready flower. If a bud forms too early, they take it out. With this approach, the grower may not need to "double stop" or even "stop" if they know (or at least guess) the flower will be at it's ideal ripeness in time for the show.
There was a time I really got into trying to figure this out. Now my show approach is to just grow a lot of flowers. Something should look good in time for the show if you have enough of them. And, if there isn't - oh-well. It's really fun to see what everyone else can do.