Note that this answer refers to the Protestant Bible. The Catholic Bible, and the Eastern Orthodox Bibles largely follow this pattern, but contain different books. More on the differences can be found here. The differences are also noted below.
First, understand that the Bible is not a single book It's a collection of 66 books, written at different time periods, by different authors (but all through the Divine Inspiration God, a phrase which has a deeper meaning, and wider implication for the Bible's trustworthiness than inspiration as you'd usually think of it.)
It is also not written in a chronological order. It is divided primarily in a topical manner.
The Bible is first divided into the Old Testament and the New Testament.
The Old Testament consists of the books that are included in the Jewish Scriptures, but the organization is different. For more info on the Jewish Scriptures, you'd be better off checking out the Judaism Stack Exchange site. I include this bit of information here because the roots of Judaism and those of Christianity are the same. It is at the boundary of the beginning of the New Testament, with the birth, life, death, and resurrection of Jesus that Christianity sprang out of Judaism. The Old Testament ends with the Jews awaiting the Messiah. The New Testament writings begin with the accounts of that Messiah - Jesus Christ.
Within the Old Testament the books are grouped into:
The Books of the Law, aka the Books of Moses, or the Pentateuch.
This consists of the first five books of the Bible: Genesis, Exodus, Leviticus, Numbers, and Deuteronomy. Collectively, this is called the Torah. These lay the foundations of the faith, introducing God as Creator, the establishment of his promise with Abraham, Isaac, and Jacob, whose line eventually becomes the Jewish people. We learn of the captivity in Egypt, the exodus, exile in the wilderness, and the giving of God's law to His chosen people.
For Jews, this represents the covenant and are the most important books period. The Sadducees believed no other part of the Bible to be inspired. These books are chronological in their subject matter, with Genesis, Exodus, and Numbers being primarily story. Leviticus represents the first iteration of the covenant, given at Sinai, in the events that occurred in the Exodus. Numbers recounts what occurs after Exodus, and Deuteronomy (literally 'Second Law') being the re-iteration of the covenant after the wandering of Israel. The names of the books are the first significant words in each book.
Next, come the Historical books, which cover the history of the Jewish people, including the establishment of their kingdom, and various (somewhat frustrating) cycles of turning away from God, punishment for their transgressions, repentance on the part of the people, and healing by God, only to have the people turn from Him and start the cycle anew. This includes Joshua through Esther.
The Catholic Bible also includes 1 and 2 Maccabees. The Orthodox Bible includes, 1, 2, 3, and 4 Maccabees, as well as 1 Esdras. Protestants consider these non-canonical, and if they are included at all, they will usually be in a separate section called "Apocrypha".
The events in these books are chronological as well, with the exception that the events of 1 & 2 Kings (also called 3rd and 4th Samuel in older traditions) are repeated in 1 & 2 Chronicles. The difference between Kings & Chronicles is that Kings focuses on the Kings, and Chronicles on the Temple.
Interestingly, in the Hebrew Bible, the "Ketu'vim" is placed after the Prophets (the Nevi'im) and thus the Hebrew Bible ends with the Temple in 2 Chronicles.
Next are the books of writings, sometimes called Poetic Writings or Books of Wisdom. This includes Job,Psalms, Proverbs, Ecclesiastes, and the Song of Solomon.
Again, the Catholic and Orthodox Bibles include extra books here: Wisdom and Sirach
These books are in no particular order - indeed both Psalms and Proverbs are merely collections and are not even chronological within themselves.
The remainder of the Old Testament is made up of prophetic books, broken up into the Major Prophets (Isaiah, Jeremiah, Lamentations, Ezekiel, Daniel) and the Minor Prophets (Hosea, Joel, Amos, Obadiah, Jonah, Micah, Nahum, Habakkuk, Zephaniah, Haggai, Zechariah, Malachi).
The Major prophets are roughly in chronological order, although there is an argument about Isaiah actually being two books.
The Minor prophets collectively are smaller than Isaiah, Jeremiah, or Ezekiel, and were often bound together in a single scroll. These books are specifically not in any chronological order, but rather are ordered the way they are based on the ending words of one book leading into the start of the next.
Within the New Testament, the books are grouped as follows:
The Four Gospels (Matthew, Mark, Luke, and John) which tell of Christ's birth, life, teachings, miracles, death, and resurrection from four different perspective.
Historically, it was believed that Matthew was the first to be written, followed by Mark, then Luke, the John. Modern scholars now often believe in what is called Marcan priority, meaning that Mark predates Matthew - both consensus solidly places John to be the last Gospel to be written. The events of the 4 Gospels are all roughly contemporaneous.
Next comes the Book of Acts (or Acts of the Apostles), which begin with a resurrected Christ, and give an overview of the birth of the Christian Church. The various missionary trips, and some basic doctrines are established in the telling of these lives.
There is very little doubt that Acts is a sequel to Luke, written by the same author, but about the events that occurred after the Gospels.
Next come the Epistles. These are letters that were written to specific Churches by various authors. There are Specific Epistles, and Generic Epistles.
First come the Specific Epistles: Romans, 1 Corinthians, 2 Corinthians, Galatians, Ephesians, Philippians, Colossians, 1 Thessalonians, 2 Thessalonians, 1 Timothy, 2 Timothy, Titus, Philemon, Hebrews. Each serves as a message to speak to an individual group of people in that exact place and time, but also apply to every people since then, including today. The Epistles are named after the group of people they were written to.
With the exception of Hebrews, these are all written by Paul (the Church erroneously attributed Hebrews to Paul) or a person claiming to be Paul. These books are ordered by length, and length alone.
Following these come the General Epistles: James, 1 Peter, 2 Peter, 1 John, 2 John, 3 John, Jude
Within each author, these are also ordered by length. 2 Peter and Jude are so similar that they are often treated as the same book.
Finally, the Book of the Revelation is the main prophetic work in the New Testament (although many of the other books contain prophecy as well.) It is largely symbolic in nature, and interpretations vary, but it speaks to the end times, and the establishment of Christ's final rule, and the creation of a new Heaven and a new Earth.