These are inscriptions written in Hebrew on broken pieces of pottery, called "ostraca". They were discovered in 1934 during excavations of Lachish, the border fortress between Judah and Egypt. There are more in the Rockefeller Museum in Jerusalem. They were found in a burnt layer, near the main gate of Lachish, which was probably burned by the Babylonian invading forces in 587/6 BC. They were written during the Babylonian invasion of Judah, which ended with the siege and destruction of Jerusalem in 587 BC, and so form a rather poignant record of the dying days of the southern kingdom of Judah.
Although some are too faint to read now, they were letters received the military governorof Lachish called Ya'osh, from a subordinate officer, called Hosha'yahu, who was leading the army in a nearby town. (Notice that both these names include the name Ya, or Yahu - the name of God).
Letter 1: (at top left)
This includes a list of Hebrew names some of which are also found in the OT, particularly in Jeremiah, although it is impossible to prove that they are the same people.
Gemaryahu cf. Gemariah (Jer 36:12)
Mattanyahu cf. Mattaniah (2 Kg 24:17)
Neriyahu cf. Neriah (Jer 32:12)
British Museum Link
Letter 2: (at top right)
This mentions the name of God (YHWH) at the right hand end of line 2 (the beginning of the line!)
To my lord Ya’osh. May Yahweh cause my lord to hear the news of peace, even now, even now. Who is your servant but a dog that my lord should remember his servant?
But there was no peace. Lachish and Jerusalem soon fell to the Babylonians, and the Jews who survived were exiled to Babylon.
Jeremiah referred to Lachish and Azekah as being the last cities to fall to the Babylonians:
Then the prophet Jeremiah spoke all these words to Zedekiah, king of Judah, in Jerusalem, when the army of the king of Babylon was fighting against Jerusalem and against all the cities of Judah that were left. Lachish and Azekah, for these were the only fortified cities of Judah that remained. (Jer 34:6-7)
British Museum Link
Another Lachish letter (number 4), in the Israel Museum in Jerusalem, mentions that the signal from Azekah can no longer be seen, probably because the city had fallen to the Babylonians.
May YHW[H] cause my [lord] to hear, this very day, tidings of good. ... And may (my lord) be apprised that we are watching for the fire signals of Lachish according to all the signs which my lord has given, because we cannot see Azekah.