LOA stands for Letter of Agreement.
Usually, for non-union engagements, this is the "contract" and it's the only official piece of paper you'll receive confirming the terms of your employment. All the basics are usually covered: fee, payment schedule, dates of engagement, dates of performance, general job duties (sometimes), indemnification and jurisdition clauses (to keep the lawyers happy). Sometimes you'll see things like supervisory titles, dates of design submittals (for designers), travel/housing, per diem, pension/retirement contributions, equipment/studio fees, or any other negotiated terms. Because a LOA is not really a standard document, they can range from one page to several pages long. The LOA is usually signed by the Artistic Director or General Director/Manager of the theatre, though it is probably processed by an administrator in your supervisory department (Production/Tech/Artistic/whatever).
A LOA does not follow any union regulations, so if you are taking a union job, a standard contract will probably follow, as dramachic explained.
Usually when a document is issued ahead of time confirming negotiated dates/rates it is called a Letter of Intent (LOI). This is an even stranger duck, as it is not really binding at all, but a nice confirmation that you will be offered employment. However, if the show is cancelled or something, this letter doesn't really offer support. If all goes well, a LOI is followed by a proper contract, be it a LOA or standard union contrat.