Deuteronomy 6: 4 "Sh'ma, Yisra'el! ADONAI Eloheinu, ADONAI echad [Hear, Isra'el! ADONAI our God, ADONAI is one]; 5 and you are to love ADONAI your God with all your heart, all your being and all your resources. 6 These words, which I am ordering you today, are to be on your heart; 7 and you are to teach them carefully to your children. You are to talk about them when you sit at home, when you are traveling on the road, when you lie down and when you get up. 8 Tie them on your hand as a sign, put them at the front of a headband around your forehead, 9 and write them on the door-frames of your house and on your gates.
Deuteronomy 11: 18 Therefore, you are to store up these words of mine in your heart and in all your being; tie them on your hand as a sign; put them at the front of a headband around your forehead; 19 teach them carefully to your children, talking about them when you sit at home, when you are traveling on the road, when you lie down and when you get up; 20 and write them on the door-frames of your house and on your gates....
A mezuzah is a piece of parchment (often contained in a decorative case) inscribed with specified Hebrew verses from the Torah (Deuteronomy 6:4-9 and 11:13-21). These verses comprise the Jewish prayer "Shema Yisrael", beginning with the phrase: "Hear, O Israel, ADONAI our God, ADONAI is One."
A mezuzah is affixed to the doorframe in Jewish homes (and the homes of Believers in YHWH/Y'shua) to fulfill the mitzvah (Biblical commandment) to inscribe the words of the Shema "on the doorposts of your house" (Deuteronomy 6:9). In that passage, YHWH commands His people to keep His words constantly in our minds and in our hearts by (among other things) writing them on the doorposts of our house.
The case containing the scroll is nailed or affixed usually (but not necessarily!) to the door frame about one third from the top of the door jamb with the top of the Mezuzah at an angle (facing toward the center of the room one is entering.) The reason many affix it at an angle is because the Rabbis couldn't decide whether to attach it horizontally or vertically. No Mezuzah is affixed at the entrance to bathrooms and toilets, for obvious reasons. (For more, please check out the article on Wikipedia.
When affixing a Mezuzah to your door one recites the prayer:
Hebrew: "Baruch attah YHWH eloheynu melekh ha-olam, asher kidshanu b'mitzvotav v'tzivanu likboa m'zuzah"
English: Blessed art Thou, O YHWH our Elohim, King of the universe, who has sanctified us by His commandments and commanded us to affix a Mezuzah.
Every time you pass through a door with a Mezuzah on it, you touch the mezuzah. This merely shows your love and respect for YHWH and His Word, and indicates that you have opened the doors of your heart to Him.
Revelation 3: 20 "Behold I have been standing at the door, and I will knock; if any man hear my voice and open the door. I will come in to him, and will dine with him and he with Me."
It is proper to remove a Mezuzah when you move, and in fact, it is usually recommended. If you leave it in place, the subsequent owner may treat it with disrespect.
The Mezuzah is significant for Believers because when we place it on our doorpost it is symbolic of us having Torah written upon our hearts (Deuteronomy 6:4-9; Proverbs 3:1,3; Jeremiah 31:31-34; Ezekiel 36:26. 27; Romans 2:14, 15; 2 Corinthians 3:3; Hebrews 8:8-12; Hebrews 10:16). In 2 Corinthians 3:3 Rav Shaul references Jeremiah 31:33 indicating YHWH has written Torah upon our hearts - therefore we are a spiritual Mezuzah since we become "containers" of YHWH's Torah!
The Mezuzah on the doorpost of our homes displays a powerful message that our household honors YHWH and that it is dedicated to His Torah.