These micro-biological organisms allow alien races to understand each other's speech. They "colonize at the base of the brain" and allow one to understand alien speech. The microbes can be injected, and go to work almost instantly. ("Premiere")
Translator microbes seem only to be required by the individual who needs to understand alien languages, not the speaker. In other words, only the listener needs to carry the microbes in order to understand alien speech, but for two-way communication to occur, both parties need the microbes. Translator microbes do not work on writing, but with the speech down, most species have no problem learning how to read in various languages.
It is notable that there is, potentially, at least one alien species, the Kalish, who cannot tolerate translator microbes. When Crichton first met Sikozu, who was a Kalish bioloid, had to learn English before she could understand what he was saying. His translator microbes allowed him to understand her.
While living with the Acquaran people, Crichton learned some phrases, in particular a greeting, in the native tongue, even though the translator microbes otherwise facilitated two-way communication without the need for Crichton to completely learn their language. ("Jeremiah Crichton") With help from John, Aeryn Sun learned English. Chiana and D'argo also learned a few English phrases as well. ("Kansas") How the microbes were bypassed in the case of Chiana and D'argo is unclear, but Aeryn learned English by watching human programming designed to improve child literacy, essentially learning through reading.
There were certain alien languages that were too complex to be translated by the microbes unless "dumbed down." Pilot's first language as well as the language spoken by Diagnosans fell into this category. The Pilots' multitasking abilities allowed them to speak in an information dense language where one word could convey numerous ideas. Most other species they might converse with were unable to understand this language without this mental capability.
NOTE: It's assumed that each character is simply speaking in their own native language, but D'Argo is often heard by the crew of Moya to speak phrases in Luxon, and Z'Haan is often heard praying in Delvian. It is expected that the translator microbes would allow these characters (and us, the viewers) to hear these words and phrases translated. Again, it's unclear how this bypass occurs.