What is a Mezuzah?

From Hebrew For Christians:

A mezuzah (מְזוּזָה) is a kosher parchment scroll (inscribed with Deuteronomy 6:4-9 and 11:13-21 on one side and the word Shaddai on the other side) that is rolled up and inserted into a decorative case. The case is then affixed to a doorway in fulfillment of the commandment given in Deuteronomy 6:9, "And you shall write them upon the doorposts (mezuzot) of your house, and upon your gates."

"Mezuzah" is the Hebrew word for "doorpost" and is a reminder that the blood of the Lamb on the doorpost identified those within as belonging to God and protected from the angel of death. The purpose of the mezuzah is to both identify those who dwell in the house as Jewish (or, for some, Messianic) and remind the dwellers of who they are and what they believe.

It has also been said that the mezuzah on the doorpost is a reminder to be mindful of what we bring through that door, of what comes into our homes. From aish.com:

If we want our internal world to reflect Godly ideals, we have to protect it against the outside world at the point of interface: the doorway. This means monitoring the contents of books, games and video that we expose our children and ourselves to.

As well, having a mezuzah on each room means that whenever we move from one domain, one sphere of activity, to another, we must renew our consciousness of God's presence and act in a way that sanctifies His Name. 

Once learned, the lesson extends beyond our home and into all areas of life. Just as a house has doorways, so too we have eyes, ears, nose, and mouth ― portals to the external world. The values of the Torah call for our mouths to eat kosher food and speak "kosher" words; for our ears not to listen to gossip; for our eyes not to run after empty desires.