Seems like there are a lot of opinions about this; it's an interesting read. Here is my take.
SOA: SOA is a services-centric architectural pattern for constructing and accessing suites of software components/services (as was noted in the answer above). SOA principles that form the SOA pattern can be found in many places, not all of which are consistent with each other, making SOA a somewhat nebulous term. A SOA service can be built with almost any modern software development technology set (see below).
API: Generally the term 'API' is used to mean the specification of how to programmatically utilize or interface with a software solution. It can refer to things such as the specifications a programming language and its components (Java APIs), the specification of how to access and/or extend a COTS solution, the specification of how to utilize a service or set of services (including the signature or data structure related to interfacing with a service), etc.
SOA and API: An API for a SOA service could include the conceptual, technology-agnostic specifications of the service (ex: one data element will be the last name of a customer), as well as the physical, technology-specific usage specifications for each physical instantiation of that SOA service (ex: two instantiations will be available, one using a JSON layout and the other an XML layout, each containing roughly equivalent physical specifications for the 'LAST NAME' data element).
Misconception: The term API should only be used to refer to RESTful or 'simple, lightweight' or JSON-based interfaces (side note: RESTful is not equivalent to 'simple' or 'lightweight'). Actually, an API can be structured following many patterns using many types of technologies, including WS* based services.
Misconception: SOA strictly follows WS* or other 'complex, heavyweight' interface approaches. Actually, SOA services can be constructed and consumed using almost any modern software development technology set, including RESTful approaches or JSON files.
More on SOA:
SOA is an architecture pattern built upon the concept that capabilities should be instantiated as services that have a crisp usage specification, and so can be utilized by any software component 'client' that can follow that usage specification, irrespective of the technologies upon which the service was developed or upon which the 'calling client' was developed. Well written services should be highly cross-compatible.
Because many open source providers and vendors have built SOA oriented frameworks that are based upon XML based approaches, notably the WS* set of standards, SOA has sometimes been mistakenly restrictively associated with WS* and/or XML related approaches. But in reality, SOA, as a conceptual pattern, is agnostic of technology. One conceptual SOA service could be instantiated once or multiple times in many forms, each instantiation choosing from a variety of technologies (XML, JSON, WS*, a REST approach, Java, C++, LAMP, ...) as determined by the designers and developers of that instantiation.
An SOA solution or solution ecosystem is one architected upon SOA principles; it is composed of services that are built and utilized following SOA principles, again, using whatever technologies the developers choose.