I see many answers explaining a Service Oriented Architecture (SOA) using even more advanced words and technical terms. I'd like to give a shot at explaining it for the layman, using an analogy in plain english.
But first a description of a SOA
SOA could be described in three layers as seen in the picture below. On one side we have the Provider and on the other side we have the Consumer, separated by a Bridge where the two sides communicate.
The consumer uses a number of Applications necessary for it's business and the provider uses Components that provide these applications with information. They communicate through a set of Services using a common architecture.
The analogy
Imagine a house on the country side, that in many ways is part of a larger community, like a city or town. The city has it's own complex systems for providing water and electricity, handling sanitation, providing transportation and other utilities. The House is the consumer in this model, the City (or community) is the provider and the pipes, sewers, powerlines, optical fibers etc. is the Infrastructure in which they communicate.
This model could loosely be compared to a SOA. The people in the house uses a number of different "applications" like radiators, computers, toilets, lamps, underfloor heating, bathtubs etc. These applications don't care how the city generates the water, creates the electricity or handles the waste as long as it works. The components of the city are generators, water pumps and sanitation areas. It provides the house with all these needs but it's up to the house to use it in what ever way it sees fit.
I hope this gave at least someone a better picture of a SOA.