The difference between 110V, 115V, and 120V, is generally just nomenclature. For various historical reasons motor ratings do not match power system ratings, but you can consider these all the same.
A phase, well without having the IEEE dictionary available, lets try this: An AC circuit will be derived from a transformer or generator winding; where there is more than one winding, the windings may be connected either wye or delta. If there are parallel or series windings, consider them combined as single equivalent windings. The number of windings powering the circuit is the number of phases present. At the system voltage, with all windings properly connected, you will measure the system voltage from each phase to all other phases. Common voltages in the US are 208 (from a wye connected set of three windings), 240 (from a delta connected set of three windings), and 480V (from either wye or delta connected set of three windings).
There may also be another conductor, called the neutral, which originates at the center of the wye, or from the mid-point of one winding. Other usable voltages are available from a phase conductor to the neutral, 120V (from any phase to neutral on a 208V wye system, or from two of the phases to neutral on a 240V delta system) or 277V (from any phase to neutral on a 480V wye system).
The one that gets a bit confusing is that when you have two phase conductors, you only have a single phase circuit; the same as when you have one phase conductor and one neutral conductor. Think of it as what is the maximum voltage in the circuit, and how many combinations of two conductors at a time provide that voltage. In any single phase circuit, there is only one combination of two conductors that provides the maximum voltage of the circuit, therefore it is single phase. In a three phase circuit, there will be three combinations that provide the maximum voltage.
120V single phase may also come from a single 240V winding with a mid-point tap. This is the most common household service voltage.
For small equipment to go into a facility, I would always just go with the available voltages. In the US, you would be hard pressed to find anyplace that has AC electric power and does not have 120V available. Beyond that "it depends".