An IGA is simply a contract between two or more governmental entities who have some sort of common authority. A typical example of an IGA in my practice is a mutual aid agreement between two fire districts. This requires each party to the IGA to provide help to one of the fire districts when it encounters an emergency that is too big for it to handle alone. Because all of the parties are fire districts, and thus authorized to respond to emergencies, they are allowed to enter into an agreement for a joint emergency response. The only meaningful difference between an IGA and any other contract is that it must be between government agencies that have some power in common. In Arizona, IGAs are governed by Arizona Revised Statutes section 11-952.
MOUs are commonly considered to be less formal than a "full" contract, but in practice they are just as enforceable as any other contract, and will be held to the same legal standards. MOUs are commonly encountered between a government agency and a private, non-profit organization such as a workers' union. They usually have few terms, and may not be expected to have any legal enforcement mechanism -- often, they are intended to be little more than an expression of common goals, without any specific action required. Despite this intention, many MOUs run more than ten pages and contain such elaborate terms that even seasoned attorneys can become confused. For this reason, I (and other government attorneys I know) prefer to avoid the term "MOU," because it can give the parties the expectation that the agreement isn't really a contract, when it actually is.