The Book of Habakkuk

Habakkuk had messages from the Lord aimed at the southern kingdom (Judah).
He prophesied that the Babylonian Empire would come to overthrow them.

Habakkuk complained to God about the wickedness of the people. Destruction and violence lay before him. The law was paralyzed, and justice never came. For the wicked surrounded the righteous and perverted justice. (Hab 1:2–4).

The Lord answered him that he was raising up the Chaldeans (Babylonian Empire).
They were dreaded and fearsome, their justice and dignity went forth from themselves. They conquered what they wanted to conquer. Their own might was their god. (Hab 1:5–11). 

The Chaldeans were Aramaic people around the capital city of the Babylonian
Empire in present-day Iraq, which freed itself from Assyria around 620 BC and crushed the capital Nineveh in 612 BC. For details see The Book of Nahum.

Habakkuk then became worried about the outcome of that answer from the Lord, what the outcome would be for Judah if the Babylonian Empire were to attack them
(Hab 1:12–2:1). The Jews saw themselves as God's people and that God wouldn't use a pagan country, probably worse than themselves, to punish them. But see 
God is sovereign in God's Attributes. He can do whatever he wants to achieve his goal, as long as it's according to his nature. God used Nebuchadnezzar,
the king of the pagan land of Babylon, to punish not only Judah, but also several
of the surrounding countries, such as Philistia, Ammon, Tyre, and Egypt. 
See The Parable of Two Eagles and a Vine.

God was true to his word and brought the Babylonian army under the king
Nebuchadnezzar to attack Judah several times. It resulted in at least these four deportations (Jer 52:28–30, 2 Kings 24:12–16): 

1. 605 BC       (The prophet Daniel was among those deported)
2. 598/597 BC (Zedekiah became the last king of Judah (2 Kings 24:17))
3. 587/586 BC (Zedekiah was taken captive to Babylon)
4. 582/581 BC

For more information, see the Babylonian captivity and Zedekiah of Judah at the end of Kings of Judah and Israel. It was the elite and the young mainly in Jerusalem who were forced to go to Babylon and it happened in several stages. The prophet Daniel was among the first to be deported (probably 605 BC) under king Jehoiakim of Judah, see Visions and Dreams by Prophet Daniel. 

The first time Nebuchadnezzar attacked Judah, he declared that Judah belonged to
Babylon and allowed a Jewish king (Jehoiachin) to rule in Jerusalem. When the Jews rebelled against him, he came and conquered them once more and let another Jewish king (Zedekiah) rule in Jerusalem. When that king also rebelled, he came once more and destroyed Jerusalem along with the temple in 587 BC. 

God allowed the Babylonians to destroy Jerusalem along with the temple, but God also promised to bring the people back from Babylon (Jer 24:6). They would return after 70 years in Babylon (Jer 25:11, 29:10, Dan 9:1–2, Isa 14:1), see 
Prophet Jeremiah. 

The Babylonian captivity lasted for 70 years until the land had enjoyed its Sabbaths
(Lev (3 Mos) 26:33–35, 2 Chron 36:21). God had commanded the nation of Israel to leave the land fallow every seven years (Sabbath year) and every 50 years
(Jubilee year), to give the land Sabbath year rest. Apparently they owed God
70 years of rest during the 442 years of their time in Israel when they didn't leave the land fallow. 442 is calculated as follows: 

• 50 years gives 7 Sabbath years and 1 Jubilee year.
• 8x50 (=400) years gives 8x7 Sabbath years and 8x1 Jubilee years.
• 8x7 + 8x1 = 64. We need 6 (70 - 64) more years.
• 6 Sabbath years gives 6x7 (=42) to be added to 400 which gives 442.

The Israelites used two calendar systems, one starting in the fall and the other in
the spring. Their calendar originally started in the fall. After the Lord brought the children of Israel out of Egypt, a change was made in their reckoning of the year so that the first month now was in the spring (Aviv (NIV) (Exo (2 Mos) 12:2, 13:3–4, 23:15, 34:18, Deu (5 Mos) 16:1), later called Nisan (Esther 3:7, Nehemiah 2:1)).
Daniel began counting the 70-year period from an autumn count. 

In Jewish reckoning, any part of a year can also be counted as a whole year. 

This means that in 606 BC would count as the first year of captivity. The end point
of the 70 years appears to be at the Feast of Tabernacles (Ezra 3:4), which was celebrated in Jerusalem in the fall just two weeks after 536 BC had started.
This two-week period was enough to extend the captivity to its 70th year: 

For details, see When Was Judah's 70-Year Babylonian Captivity?

God would bring the Babylonian Empire against the people of Judah, to discipline them in their rebellion and sin. But afterwards God would judge the
Babylonian Empire (Hab 2:4–20). 

Note that Habakkuk proclaims that the righteous shall live by his faith (Hab 2:4b),
see Obedience Through Faith in The Book of Romans.

After the death of king Nebuchadnezzar, the Babylonian Empire began to collapse. 
It was conquered by the Medes and Persians in 539 BC (Dan 5:24–31). 
Cyrus the Great of Persia took power. He proclaimed in 537 BC that the descendants of those deported from the kingdom of Judah would be allowed to return and rebuild the temple (2 Chron 36:22–23, Ezra 1:1–4, Psalm 106:44–47), see
The Book of Ezra. The returning Jews rebuilt the temple in Jerusalem under 
Zerubbabel, see The Book of Haggai.

See also The Prophet Books in the Old Testament.