I wonder about OP's teacher... Be that as it may, the explanation is fairly simple. Here's the Hebrew and English for Ezekiel 9:4
וַיֹּ֤אמֶר יְהוָה֙ [אלו] אֵלָ֔יו* עֲבֹר֙ בְּת֣וֹךְ הָעִ֔יר בְּת֖וֹךְ יְרֽוּשָׁלִָ֑ם וְהִתְוִ֨יתָ תָּ֜ו עַל־מִצְח֣וֹת הָאֲנָשִׁ֗ים הַנֶּֽאֱנָחִים֙ וְהַנֶּ֣אֱנָקִ֔ים עַ֚ל כָּל־הַתּ֣וֹעֵב֔וֹת הַֽנַּעֲשׂ֖וֹת בְּתוֹכָֽהּ׃
wayyōmɛr YHWH ʾēlå̄w ʿăḇōr bəṯōḵ hå̄ʿīr bəṯōḵ yərūšå̄lå̄yim wəhiṯwīṯå̄ tå̄w ʿal-miṣḥōṯ hå̄ʾănå̄šīm hannɛʾɛ̆nå̄ḥīm wəhannɛʾɛ̆nå̄qīm ʿal kål-hattōʿēḇōṯ hannaʿăśōṯ bəṯōḵå̄h
The Lord said to him, “Go through the midst of the city, even through the midst of Jerusalem, and put a mark on the foreheads of the men who sigh and groan over all the abominations which are being committed in its midst.” NASB
I've put the key bit in bold: it says (fairly) literally "put a taw on the foreheads...".
So that's why the teacher said it was a
ת = taw = last letter of the Hebrew alephbet -- because that's the instruction that's given in Ezekiel 9:4!
But why an "X" or a "+"? Because that's the letter-shape of the letter taw in Ezekiel's time, in the so-called "paleo Hebrew" alphabet. You can see the shape well in this "letter" (ostracon) from Lachish (this is #2):
The letter taw has been circled in red (I don't think I've missed any).
So in answer to OP's question:
What was the mark and why that character?
The mark really was a paleo-Hebrew taw, because that's what the LORD instructed the the "man clothed in linen" to do.
† As was pointed out to me, a properly literal translation would be something like: "Go through the midst of the city, even through the midst of Jerusalem, and taw a taw on the foreheads...", since the verb here, for "put a mark" is
I תָּוָה, see BDB, p. 1063a for more.