There are many kinds and forms of breach of contract under New York law; it isn't one size fits all, and therefore, it should come as no surprise that there correspondingly, a very wide variety of rules that govern these cases.
Some agreements must, by necessity, be reduced to writing in order to be enforceable pursuant to General Obligations Law 5-701, which is more commonly known as the Statute of Frauds. Some examples of contracts that fall under this statute include agreements pertaining to the purchase of real estate, or agreements that cannot, by their terms, be performed within one year.
Other contracts, such as for goods, are typically governed by the Uniform Commercial Code, which will dictate, in large part, what each side's remedies and damages are in the event of a breach of the agreement.
Employment contracts have their own set of laws as well, as they may implicate collective bargaining agreements (such as with unions), contain conditions precedent to commencing suit. They may also dictate that any and all disputes take place before a particular judicial forum, such as arbitration, or which jurisdiction's laws will govern the dispute.
In addition, and particularly in the non-compete realm, some agreements may inherently be unenforceable as a matter of law on public policy grounds.
As a general rule, it is probably a good idea to consult with an attorney knowledgeable in this area rather than making any assumptions about your potential breach of contract matter.