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OSGi is really an adjective. It generically refers to the OSGi Alliance, an industry organization that creates specifications. In particular, the OSGi Service Platform Specification R4. See Wikipedia for more details and pointers.
Why should you care?
Well, since Eclipse 3.0, Equinox, the Eclipse runtime, has been based on this specification. Actually Eclipse 3.0 and 3.1 were based on OSGi Service Platform Specification R3 (whew, that's a mouthful) plus some extensions. OSGi R4 (much simpler to say) includes many of these extensions and Eclipse 3.1.2 and 3.2 are need only a modest few extensions.
The Equinox project includes implementations of all of the optional and mandatory R4 core elements and many of the optional services. Many of these are used by the OSGi Alliance as the Reference Implementations for the R4 specification.