The Bible was originally written in three ancient languages: Hebrew, Aramaic, and Greek

Hebrew

Most Old Testament books: Genesis through Malachi (the 39 books of the Protestant Old Testament) were primarily written in Biblical Hebrew.
Details: This was the primary language of the ancient Israelites. 

Aramaic

Book of Daniel: Chapters 2:4 through 7:28 are written in Aramaic.

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Book of Ezra: Chapters 4:8 through 6:18 and 7:12 through 7:26 are written in Aramaic.

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Book of Jeremiah: Chapter 10, verse 11 is a single sentence written in Aramaic.

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Book of Genesis: Two words (Witness Heap / Galeed) appear in Aramaic in Genesis 31:47. 

Greek

All New Testament books: Matthew through Revelation (all 27 books) were originally written in Koine (common) Greek.
Deuterocanonical / Apocryphal books: Certain books found in Catholic and Orthodox Bibles (such as Tobit, Judith, 1 and 2 Maccabees, Wisdom of Solomon, Sirach, and Baruch) were also originally composed or preserved in Greek.