On Earth:
The primary language (most residents secondary language after their native tongue) is English. Most residents speak their native tongue as well. These native tongues are determined by geography primarily, though some also inherit theirs through cultural associations.

Anyone who graduated from the Specials Academy speaks their native tongue, Japanese, French, English, and at least one other elective.

The upperclass/politicians of Earth speak French as the language of politics, English as the language of the commons, and their native language at least.

Furthermore, many areas (particularly eastern ones, and areas that have spaceports to the Colonies) use Japanese as a language for trade and tourism, though it is not always spoken in the home enviroment (with the exception of Japan). Think of the way English is used today in the United States: everyone speaks it practically, but there are many families that speak their "native" language in the home enviroment, and speak it more fluently. Japanese loanwords have also crept into many languages over the years; the most noticeable example of this is the use of familial terms such as "neesan" or "oniisama" to refer to siblings.

 
On the Colonies:
The first language of the colonies (L1 was colonized first) is Japanese, so it can be assumed that all colonists speak Japanese as the "national language". Each colony has its own language based on the major origin of their settlers (Japanese, English, French, Arabic and Chinese respectively).

The colonies also have developed there own language, a colonial dialect known as Scat, which is a combination of Japanese, English, French, and Arabic. This language evolved during the time when only four colonies were in space, and was used as a form of "equal communications", since not all immigrants spoke Japanese during the first few generations. Scat is still used by the lower end of society, particularly on the streets of L2, in C side.

Finally, on the streets of the Breaks there is yet another language used on the streets among the assassins and other less reputable residents. Based on American sign language, it is used for negotiation, assassination, and smuggling. It is known as Assassin's Sign.

 
Special Language Notes:
Cinq language is a dialect of French known as Cinqant. The relation between "pure" French and Cinquant is similar to that of American English to True (British) English, with Cinquant being the more formal of the two.

 
Who Speaks What:

HEERO YUY: Heero is fluent in English, Mandarin and Japanese. English is almost universally accepted as the programming language for computers, and Japanese is the native language for L1. He also speaks the Assassin's Sign.

DUO MAXWELL: Duo is a master of languages. He has a knack for picking them up very naturally, though he has trouble writing them, and tends to speak informally or with outright rudeness. At last count he spoke English, French, Spanish, Japanese, German, Mandarin, Arabic, Russian, Assassin's Sign and Scat.

TROWA BARTON: Trowa is fluent in English, Japanese and French. French is his native language (both of L3 and of the circus). He speaks a little bit of most European languages for traveling purposes. While it has been implied he understands what Wufei says when Wufei speaks Mandarin, this is a misperception- he is only able to get the jist, rather then the true meaning. Due to his time among the dregs of the Colonies, he also speaks Scat and assassin's sign.

QUATRE RABERBA WINNER: As part of the extreme upper class, Quatre has had all of the best tutors. His language skills, in contrast to Duo, are very polite and formal. Japanese, Arabic, Farsi, English, and French/Cinquant are his main languages, though he understands bits and pieces of about ten other tongues.

CHANG WUFEI: Wufei speaks Mandarin, Cantonese, Japanese and English. He rarely speaks anything aside from Mandarin, only condescending to use Japanese when the other person does not speak his native tongue. As a scholar, he learned to read Latin, which gives him a basic understanding of many of the Romantic languages.

ZECHS MERQUISE/MILLIARD PEACECRAFT: Milliard, as part of the European nobility of Cinq, speaks Cinquant, as well as Japanese, English, and French. At the Academy, his elective language was Russian.

TREIZE KHUSHRENADA: Treize spoke English, German, French, Spanish, and Japanese. He also spoke Russian passably, because of the Russian branch of his family on his grandfather's side. His elective language was Arabic.

RELENA DARLIAN PEACECRAFT: Relena speaks French and English, and has been learning Cinquant and Japanese.

HILDE SCHBEIKER: Hilde is a second-generation German immigrant, and as such, speaks English, German, and Japanese.

CATHERINE BLOOM: Catherine speaks French, English and bits of many other languages, including Japanese, which she learned while touring the Colonies.

DOROTHY CATALONIA: Dorothy speaks Spanish, French and English. She holds a disdain for learning languages (she much prefers government and economics), and relies on translators when necessary.

SALLY PO: Sally speaks English, Mandarin, basic Japanese, and some French. She did not graduate from the Specials Academy, so does not have the same criteria as other former military personal in SNK.

LUCREZIA NOIN: Noin speaks Italian, English, French, and Japanese. Her elective language was Spanish.

LADY UNE: Une speaks French, German, English and Japanese. Her elective language was Cantonese.

DARKFLIGHT: Darkflight speaks Japanese, Scat and Assassin's Sign.

ATSUKI: Atsuki speaks Japanese, Arabic, English, French, and the bits of Assassin's Sign related to her profession.

ILENE KEETS, CHRISTOPHER JOHNSEN AND HELENA ROSEMBAUM: All speak English as their native languages and one elective. Chris and Helena's elective is French, while Ilene pursues Italian (primarily due to her love of music).

MATSUURA SHINOBU: Shinobu speaks basic English, Japanese, Scat, and Assassin's sign. Due to the fact that Cliffside considers him an exchange student, he is allowed to take English as his elective language.