"Local Amendments" to the NEC - What is meant by this term?
Building codes (including the NEC and many others covering a variety of subjects) are usually written by independent organizations who specialize in the subject.

Local laws are passed by governments. Since local politicians are generally not electricians / structural engineers / fire marshals etc, they "adopt" the 3rd party codes by requiring all local construction to comply with them. But sometimes for various reasons the government wants to make additional requirements or changes, e.g. to take into account local conditions, additional safety, historical reasons, etc. They can change the code in any way they want, maybe prohibiting something they feel is risky or allowing something that would normally be prohibited. Or maybe they just change the wording or definitions a bit to clarify or be consistent with other local laws.

In the USA, building codes are often selected by state government, although local cities and counties may have additional rules they require. Note that every state has their own building codes, which is why you must look up the rules for your specific location. Even states that adopt the NEC (which is most of them, but not all) may use different versions. E.g. here is the current NEC adoption, as of September 1 2015:



The changes themselves can be minor and pedantic or major. For an example, here is the Massachusetts electrical code. Note that it starts by saying the code is the 2014 National Electrical Code, with a number of changes.