Watches of the night are mentioned in both the Old and New Testaments.  What are they?

Old Testament watches of the night

In Old Testament times the night was divided into three watches: beginning, middle, and morning.

The first of these watches is mentioned in Lamentations 2:19.

Lam 2:19  “Arise, cry out in the night, At the beginning of the watches; Pour out your heart like water before the face of the Lord. Lift your hands toward Him For the life of your young children, Who faint from hunger at the head of every street.”

The middle watch is mentioned in Judges 7:19.

Jdg 7:19  So Gideon and the hundred men who were with him came to the outpost of the camp at the beginning of the middle watch, just as they had posted the watch; and they blew the trumpets and broke the pitchers that were in their hands.

The morning watch, (the last watch), is mentioned in Exodus 14:24 and 1 Samuel 11:11.

Exo 14:24  Now it came to pass, in the morning watch, that the LORD looked down upon the army of the Egyptians through the pillar of fire and cloud, and He troubled the army of the Egyptians.

1Sa 11:11  So it was, on the next day, that Saul put the people in three companies; and they came into the midst of the camp in the morning watch, and killed Ammonites until the heat of the day. And it happened that those who survived were scattered, so that no two of them were left together.

Presumably, these watches were of roughly equal length, about four hours long.  The book of Psalms also mentions night watches.

Psa 63:6  When I remember You on my bed, I meditate on You in the night watches.

Psa 90:4  For a thousand years in Your sight Are like yesterday when it is past, And like a watch in the night.

Psa 119:148  My eyes are awake through the night watches, That I may meditate on Your word.

New Testament watches of the night

By the time of the New Testament, a fourth watch had been added (presumably as a result of Roman influence) so that the night was divided into four watches of about three hours each.  They seem to have been called the first, second, third, and fourth watch.  The second, third, and fourth watches are mentioned directly.

Mat 14:25  Now in the fourth watch of the night Jesus went to them, walking on the sea.

Mar 6:48  Then He saw them straining at rowing, for the wind was against them. Now about the fourth watch of the night He came to them, walking on the sea, and would have passed them by.

The fourth watch runs from about 3:00am to 6:00am.

Luk 12:38  And if he should come in the second watch, or come in the third watch, and find them so, blessed are those servants.

Interesting, Jesus made a possible allusion to these four watches when He spoke of His return.

Mar 13:34  It is like a man going to a far country, who left his house and gave authority to his servants, and to each his work, and commanded the doorkeeper to watch.
Mar 13:35  Watch therefore, for you do not know when the master of the house is coming—in the evening, at midnight, at the crowing of the rooster, or in the morning—
Mar 13:36  lest, coming suddenly, he find you sleeping.

The four times of day Jesus mentioned fit with the ends of the four watches:
1. in the evening fits with the end of the first watch (about 9:00pm)
2. at midnight fits with the end of the second watch (midnight)
3. at the crowing of the rooster fits with the end of the third watch (about 3:00am).  Roosters can start at this hour
4. in the morning fits with the end of the fourth watch (about 6:00am)
 
Conclusion
Watches of the night in Old Testament times were three divisions of the night of about four hours each.
 
Watches of the night in New Testament times were four divisions of the night of about three hours each.