Emanuel Tov, Textual Criticism of the Hebrew Bible (Minneapolis: Fortress, 2001) 218-219.
When you open an Old Testament/Hebrew Bible in Hebrew (say, a BHS), the letters' script that you see is known in rabbinic sources as the Assyrian "square" script, and among many scholars as the Aramaic script. That's also the script that you'll see in synagogues. But the Hebrew Bible wasn't always written in that. As Tov states, "Originally, the biblical books were written in the 'early' Hebrew script which developed from the proto-Canaanite script in the tenth or ninth centuries B.C.E." You can see this paleo-Hebrew script when you watch the Ten Commandments, for the commandments on the two tablets are in paleo-Hebrew. Also, Samaritan Pentateuchs that people get in Israel are in the paleo-Hebrew script. To see what the alphabet looks like in paleo-Hebrew, see Paleo-Hebrew alphabet - Wikipedia, the free encyclopedia.
According to Tov, the transition from the Hebrew to the Aramaic script occurred gradually during the Second Temple Period, which encompasses the time from Ezra (fifth century B.C.E.) to the destruction of the Jerusalem temple in 70 C.E. My guess is that the Jews adopted the Aramaic script because Aramaic was the lingua franca of their day. Why did the rabbis call it the Assyrian script, rather than the Aramaic script, or (since the rabbinic B.T. Sanhedrin 21b says Ezra brought it from the Babylon captivity) the Babylonian script? Here's the answer that Tov gives, though I can't say I understand it: "...it is called the 'Assyrian script' due to the fact that its ancestor, the Aramaic script, was in use in the Assyrian empire." In the biblical story about Hezekiah and the Assyrian King Sennacherib's attempted conquest of Jerusalem, Hezekiah's officials know that the Assyrians can speak Aramaic (II Kings 18:26; Isaiah 36:11). Maybe rabbinic Judaism associated Aramaic with Assyria because Assyria was the first big-time conqueror of Israel, so it loomed large in Jewish memory.
There are rabbinic references that say God originally gave the Torah in paleo-Hebrew, but Ezra introduced the idea of conveying it in the Aramaic script. Israel then "selected the Assyrian script and the Hebrew language, leaving the Hebrew characters and Aramaic language for the [ordinary people]" (B.T. Sanhedrin 21b). Tov also refers the reader to Origen, Epiphanius, and Jerome for this point.
But B.T. Sanhedrin 22a presents another point of view: God originally gave the Torah in the Aramaic script, but Israel sinned, so the beautiful Aramaic script became the broken paleo-Hebrew script. The idea that the Aramaic script was God's chosen script for the Torah may explain why there are elements of Judaism that seek deep significance in the shapes of the Aramaic letters. See Yosef's comment under my post, Category or Spiritual Path?.