What is Engagement?

Legal Definition
An engagement, betrothal, or fiancer is a promise to wed, and also the period of time between a marriage proposal and a marriage. During this period, a couple is said to be betrothed, intended, affianced, engaged to be married, or simply engaged. Future brides and grooms may be called the betrothed, a wife-to-be or husband-to-be, fiancée or fiancé, respectively (from the French word fiancer). The duration of the courtship varies vastly, and is largely dependent on cultural norms or upon the agreement of the parties involved.

Long engagements were once common in formal arranged marriages, and it was not uncommon for parents betrothing children to arrange marriages many years before the engaged couple were old enough.
-- Wikipedia
Legal Definition
This word is frequently used in the French law to signify not only a contract, but the obligations arising from a quasi contract. The terms obligations (q. v.) and engagements, are said to be synonymous 17 Toull. n. 1; but the Code seems specially to apply the term engagement to those obligations which the law, imposes on a man without the intervention of any contract, either on the part of the obligor or the obligee. Art. 1370.
-- Bouviers Law Dictionary
Legal Definition
In French law. A contract. The obligation arising from a quasi contract The terms "obligation" and "engagement" are said to be synonymous, (17 Toullier, no. 1;) but the Code seems specially to apply the term "engagement" to those obligations which the law imposes on a man without the intervention of any contract, either on the part of the obligor or the obligee, (article 1370.) An engagement to do or omit to do something amounts to a promise. Rue v. Rue, 21 N. J. Law, 369. In English practice. The term has been appropriated to denote a contract entered into by a married woman with the intention of binding or charging her separate estate, or, with stricter accuracy, a promise which in the case of a person sui juris would be a contract, but in the case of a married woman is not a contract, because she cannot bind herself personally, even in equity. Her engagements, therefore, merely operate as dispositions or appointments pro tanto of her separate estate. Sweet.
-- Black's Law Dictionary
Legal Definition
An agreement; a contract.
-- Ballentine's Law Dictionary