Once converts are established on a world, they will continue to grow, by about 10% each turn. (Each convert has a 10% chance of converting another population, each turn.) If the converts ever reach 100% of the population, then the Missionary will automatically capture the world. And even before then, the converts act as spies: the Missionary that owns the converts will get a full report on the activities at any world with even one convert.
You can attempt to eliminate converts by unloading consumer goods at the world (or invite a Trader ally to do it for you). Each consumer good unloaded has a 50% chance of unconverting one convert, so if you have a lot of converts, this can get expensive.
You can also fire at converts, killing them. This is much more efficient then buying them back, but it has a cost too. Firstly, unless you are a Positronix (or another Missionary pronouncing Judgment), you will lose points for killing innocent civilians. Secondly, if the Missionary in question has not yet pronounced Judgment, then you will also be making their converts into martyrs, awarding them additional points. Finally, unless you have a substantial non-convert population, you may not have enough population left after killing the converts to work your industry or mines (converts are good workers).
Finally, you can try to get rid of converts simply by migrating them to a neighboring world. Of course, this will just put more converts at the neighboring world.
Only one Missionary at a time can own the converts at a given world. If a second Missionary attempts to make converts at a world, all of the converts will be owned by either one Missionary or the other (determined by random chance, and weighted on the relative number of converts for each Missionary).
In the RSW Client, the number of converts of a population, and the owner of the converts, are indicated on the World Detail panel. This information also appears on the text-based turn report accessible from the web page.