What determines whether a unit is a base unit? I've had base units explained to me as units that cannot be broken down into a combination of other units. The Ampere is considered a base unit, however it can be written as coulombs per second.
Is there a better explanation of SI base units?
 A: The base units are simply listed in each system of units. In the International System of Units (SI), there are seven base units: kilogram, metre, candela, second, ampere, kelvin, and mole. 
I copied the previous sentence from Wikipedia.
Note that they are some – in this case seven – units that are enough to create other units by considering products of powers of the base units (which include various ratios etc.).
Most of the base units are "named" by words that don't contain any prefixes. In the SI system of units, the kilogram is an exception and it's considered the base unit instead of one "gram" even though one "kilogram" sounds like a derived word with a prefix while one "gram" unit sounds as a base one. It's just a convention – in this case, one kilogram was picked as the "base unit" because there existed and still exists the "prototype kilogram" made from platinum and iridium in France.
Similarly, in the 19th century engineering (and electric engineering) "CGS" units, the base units were centimeters, gram, and second. The "gram" was the linguistically simpler one but instead, the "centimeter" was considered more elementary than a "meter". Just another convention.
One can have other systems based on Imperial units or e.g. "natural units" in which the universal constants of Nature, such as the speed of light and Planck's constant, are set equal to one or some other simple numerical constants.
