Bosnian is a variant of Serbo-Croatian. It is the official language in Bosnia and Herzegovina as well as Montenegro and a native language of a little over 2 million people.
The Federation of Bosnia and Herzegovina officially has three languages: Bosnian, Croatian and Serb. All three languages are mutually understandable. For various reasons, Bosnian is in wide use throughout the Balkans. The Serbo-Croatian concept, as well as the separate variants of the language (Serb, Croatian, Bosnian), was in fact based on the most wide-spread dialect in the area, the Shtokavian one from Eastern Herzegovina.

Bosniak is (not) Bosnian

It is not uncommon for Bosnian being also referred to as Bosniak. This is actually one of many things considered controversial on the Balkan Peninsula – while Bosnians insist the only correct name for their language is Bosnian, the Croats and Serbs insist on Bosniak being used to refer to the language within the country of Bosnia and Herzegovina and Bosnian as a term to include also Bosnian Croats and Serbs living outside of Bosnia and Herzegovina. It went even as far as Serbs refering to the Bosnian language as "the Language spoken by Bosniaks" in official documents, only to not having to recognize the language at all. Symbols and details really do matter in these lands, and based on previous experience, it won't be easy to overcome these disputes (mainly if the tensions are being deliberately encouaged, as seen in latest efforts in destabilizing Bosnian society through stirring debate about the Republika Srpska national day in 2018, etc.).
Internationally and within the language and translation industry, as well as here at idioma, the recognized name is Bosnian, though.

Not interested elites, no codified language

Since the 1990s, Bosnian has developed considerably, integrating literary traditions from the 20th century and adopting loan words from the Islamic and Oriental worlds. Arabic, Persian and Ottoman words differentiate the Bosnian version of Serbo-Croatian vocabuly from its siblings noticably, due to the religious ties with the Islamic world. In fact, due to this close bond of the Bosnian elite to Oriental cultures and lack of true emancipation of the Bosnians, the language failed to be codified in the 19th century, unlike Serbian and Croatian.

Alphabet

Bosnian originally used the Cyrillic alphabet, but today also the Latin alphabet is in use due to the influence of Serbo-Croatian when Bosnia was part of the Socialist Federal Republic of Yugoslavia. Both alphabets are equal, but Cyrillic is used increasingly less today. Cyrillic, however, has greatly influenced the development of the Bosnian language and remains a link to the past. Latin
A B C Č Ć D Dž ĐE F G H I J K Lj M N Nj O P R S Š T U V Z Ž
a b c č ć d dž đ e f g h I j k l lj m n nj o p r s š t u v z ž

Cyrillic
А Б Ц Ч Ћ Д Џ Ђ Е Ф Г Х И Ј К Л Љ М Н Њ О П Р С Ш Т У В З Ж
а б ц ч ћ д џ ђ е ф г х и ј к л љ м н њ о п р с ш т у в з ж