An arch is a vertical curved structure that spans an elevated space and may or may not support the weight above it, or in case of a horizontal arch like an arch dam, the hydrostatic pressure against it.
Arches may be synonymous with vaults, but a vault may be distinguished as a continuous arch forming a roof. Arches appeared as early as the 2nd millennium BC in Mesopotamian brick architecture, and their systematic use started with the ancient Romans, who were the first to apply the technique to a wide range of structures.
An arch carries load by compression. It is helpful to think of the arch as the opposite of a cable, because a cable carries its load by tension.
Arches have an additional complication when compared to cables. Because the arch is in compression, the arch wants to buckle. For this reason, an arch can never be as slender as a cable. To prevent buckling an arch needs to have some bending stiffness. Because an arch has bending stiffness it cannot change shape to suit the load, as a cable does.
For a given load there is one particular shape of arch which would carry that load by pure compression in the arch, and this is the most efficient arch shape for that load. This shape is the funicular shape. The easiest way to find the funicular shape is to put the load on a cable, and see what shape the cable takes up (this will be pure tension). Flip that shape upside down and you have the shape of an arch that carries the load by pure compression.
When the shape of an arch is not the correct funicular shape for a given load, the arch will carry the load by a combination of compression and bending moment. The further away the shape is from the funicular shape, the more bending moment there will be in the arch.
This video discusses how the different structural forms we have discussed so far can be used to build bridges in different situations.