Simply said, OSGi is a dynamic module system for Java. It defines means to install, uninstall, update, start and stop modules. Those modules are called bundles, but are, in their simplest form, actually Java jar files with a special Manifest. Modules can be installed, uninstalled etc without stopping or restarting the Java VM.
An OSGi framework manages the described lifecycle of and dependencies between the bundles in a secure way. A bundle needs to state which Java packages it exports and which it imports. The import and export statements can be annotated with version information, so that you even can have more than one version of the same package in the same Java VM.
The OSGi Alliance is the organization that specifies the OSGi framework and many accompanying services, e.g. for managing configuration data, device access, etc.
This is just a very basic overview. OSGi is much more.
Please have a look at https://www.osgi.org/resources/architecture/ (an introduction to OSGi’s architecture) and https://www.osgi.org/resources/where-to-start/ (a lot of links and further readings recommended by the OSGi Alliance).
Update 07/2021
As of 2021 the OSGi Alliance moved all resources and work to the Eclipse Foundation. It continuous its work in the newly established OSGi Working Group.