If someone doesn't pay their rent by the time it is due you can go to the courthouse and fill out the paperwork to get a judgement of non-payment. Usually the next week you get a court date to actually get that judgement from the court. A week after getting that judgement you can begin the eviction process which might take a couple days or a couple weeks depending on the availability of the deputy whose in charge of overseeing evictions.
I think what sockgirl77 is referring to, is that you can't go through with the eviction if they pay the amount due before you finish the process (that means all of their possessions out of the premises). If, however, you go through the process and get three judgments against them within a year, you can choose not to accept their payment and evict them anyway the third time around.
In theory you can have someone evicted within 3 weeks if the timing falls just right.
Atleast, that is how it used to work in Maryland. If it has changed in the last 10 years I could be completely wrong.
