Some people are under the perception that Hebrew changed from the time of Moses to the time of the bible we have today. They seem to think that ancient Hebrew is not the same as Hebrew today.
"Paleo-Hebrew", also known as K'tav Ivri in Hebrew was a different font, not a different aleph bet. . . Think of fonts in English: do any of them change the meaning of the letter? Nope – and neither does the font in Paleo-Hebrew. The letters in the ancient font (shown at the bottom of the image) are the same letters in the modern Hebrew aleph bet. All of the scripts in the picture represent the same exact letters, sounds, everything. They look different, just as English cursive handwriting letters look different than block English letters, but they are the same letters.
Which came first, k'tav Ivri or k'tav Ashuri? Most people think that k'tav Ivri (the bottom script in the image), aka "proto" or "paleo" Hebrew was the original form of letters for Hebrew, and was used by Moses to write the original thirteen Torah scrolls. Others contend that k'tav Ashuri (the top font, the one used in Torah scrolls) was in use by Moses, but as time went by it fell into disuse in favor of k'tavi Ivri. Whichever was ancient, about 2500 years ago the prophet עזרא / Ezra, returning from Babylonian exile, decided that k'tav Ashuri would be the only font used from that point on for a Torah scroll.
עזרא הסופר / Ezra the Scribe (a Jewish priest) was an amazing man. The Talmud states: "the Torah could have been given to Israel through Ezra, if not that Moses preceded him" (Sanhedrin 21b). Quite a compliment indeed!
Moses hand wrote thirteen Torah scrolls -- one for each tribe and one for the אָרוֹן הַבְּרִית /ʾĀrôn Ha'brît / ark of the covenant. When Jews were allowed to return from Babylonian exile it was עזרא הסופר / Ezra the Scribe who discovered three of those original hand written Torah scrolls penned by Moses himself.
Ezra found a few minor variances in the three scrolls. One Torah had the word נערי written as זאטוטי (Aramaic). A second difference was that in one the word מעונה was written מעון without the final ה. The last difference was that the word היא was written הוא (but pronounced the same -- 'hi'). Other than these, the texts were identical -- so there was no difference at all that impacted a single word or meaning. . .
Ezra also helped to found כְּנֶסֶת הַגְּדוֹלָה / Anshei Knesses Hagdolah / The Great Assembly. This august Sanhedrin included the last of the prophets including Mordecai (of Queen Esther fame), Daniel, Ezra, Nehemiah, Haggai, Zachariah, Malachi, and Shimon HaTzadik (Simon the Righteous). The Great Assembly codified the Jewish bible (to include the books of the Prophets and Writings). Ezra himself wrote the books of Ezra, Nehemiah and Chronicles. Some missionaries think the bible is all "equal" and this is not true. The Torah (תּוֹרָה / instructions) is the holiest part of the T'nach, and the mitzvot in it cannot be changed (added to or subtracted from). The books in נְבִיאִים / Nevi'im / Prophets give us much of our history, along with messages that are important to us, the future generations. The messages of the prophets re-enforced the mitzvot of the Torah. Lastly, the books in כְּתוּבִים / Ketuvim / Writings are inspired by G-d, but they are not as holy as prophecy -- they do not rise to that level. Still, they have important messages, poetry, stories, etc.
For modern Hebrew readers reading a printed Chumash or T'nach (such as the Artscroll Stone Edition, or the Judaica Press translation) in Hebrew are reading the "Printed" Hebrew font.
A Torah scroll is handwritten using extensive safeguards using the Ashuri font (k'tav Ashuri).
Most handwritten notes, letters, and such are written in Hebrew cursive, and proto / paleo Hebrew used the Ivri (Hebrew) font known as k'tav Ivri.
The Hebrew word עברי (ivri) becomes "Hebrew" in English. It can be a noun and an adjective. The word stems from עֵבֶר / Ever, the great-grandson of שֵׁם / Shem (son of נֹ֫חַ / Noah and great-great-great-great-grandfather to אַבְרָהָם / Abraham).
To further confuse the issue missionaries will often speak of the "Masoretic Text" (MT) as if it is the Hebrew of the T'nach. It is not. Hebrew is normally written using only consonants. Vowels are not part of written Hebrew. If you were to visit Israel and pick up a local newspaper you would only see Hebrew consonants. Fluent Hebrew readers do not use vowel notations at all, even today. Torahs (of course) do not USE cantillation, e.g. the Masoretic vowel markings.
In the 9th century CE Jewish scribes who copied the T'nach added vowels and punctuation marks for the first time. Hebrew is written with consonants only, and the vowels are inserted by the reader when it is read out loud. It is usually pretty easy to know how to pronounce words based on the letters in them so it isn't as confusing as it might seem to a non-Hebrew speaker.
The idea of punctuation marks were to help the person less familiar with Hebrew. The סופרים / Sopherim (scribes who are responsible for writing Torahs according to the 20+ rules for copying them) wrote the consonantal text. The scribes also added the marginal notes. If you see dots or dashes under Hebrew letters you are seeing the Masoretic punctuation marks known as cantillation. Do not let missionaries confuse you with deflections into arguments about the Masoretic Text -- it is a red herring.