In Matthew we read:
Matt 1:18 Now the birth of Yeshua Messiah (Jesus Christ) was on this wise: When as his mother Miriam (Mary) was engaged (the engagement is kiddushin) to Joseph, before they came together, she was found with child of the Holy Spirit.
Kiddushin (the engagement) is sanctification or dedication.
"Sanctification," indicates that what is happening is not just a social arrangement or contractual agreement, but a spiritual bonding and the fulfillment of a mitzvah, a Divine precept. "Dedication," indicates that the couple now have an exclusive relationship, that involves total dedication of the bride and groom to each other..." (http://ohr.edu/yhiy/article.php/1087)
The word "kiddushin" is from the word "kadosh" (holy).
Term "kidushin" is term of the sages in Mishna and Gemara (Talmud). In Torah there is the other word of "laaros'' (to betroth). An engaged girl is "naara meoreset". (http://www.machanaim.org/yearroun/dach_brak/lek6.htm)
In Torah we read about the engagement:
Deut 20:7 And what man [is there] that hath betrothed a wife, and hath not taken her? let him go and return unto his house, lest he die in the battle, and another man take her.
The mutual relations between Hashem and Israel are the engagement too.
Hos 2:19 And I will betroth thee unto Me for ever;
yea, I will betroth thee unto Me in righteousness, and in judgment, and in
lovingkindness, and in mercies.
Hos 2:20 I will even betroth thee unto Me in faithfulness: and thou shalt
know the LORD.
Jer 2:2 Go and cry in the ears of Jerusalem, saying, Thus saith the LORD; I remember thee, the kindness of thy youth, the love of thine espousals, when thou wentest after Me in the wilderness, in a land [that was] not sown.
There are two stages in the Jewish marriage: kiddushin and nisu'in (full-fledged marriage). During kiddushin the bridegroom gives a ring to his bride. "The nisu'in is accomplished through "chupah" - the husband uniting with the wife under one roof for the sake of marriage." (http://www.chabad.org/library/article_cdo/aid/477321/jewish/Kiddushin-Betrothal.htm)
"The nisuin (from a word meaning elevation) completes the process of marriage. The husband brings the wife into his home and they begin their married life together.
In the past, the kiddushin and nisuin would routinely occur as much as a year apart. During that time, the husband would prepare a home for the new family. There was always a risk that during this long period of separation, the woman would discover that she wanted to marry another man, or the man would disappear, leaving the woman in the awkward state of being married but without a husband. Today, the two ceremonies are normally performed together." (http://www.mechon-mamre.org/jewfaq/marriage.htm)