TLDR: Apache Karaf is much more 'batteries-included'. It can also run on any OSGI runtime.
Apache Felix (which is an implementation of the OSGi 4.2 framework)
Sort of. Apache Karaf can use Apache Felix. Apache Karaf can also use Equinox or another OSGi runtime. By default, the Apache Karaf standard download does come with Apache Felix.
Apache Aries (which is an implementation of the Blueprint standard)
Again, sort of. The standard download of Apache Karaf does come with Apache Aries by default.
Now let's talk about what OSGI (Apache Felix, Equinox) is and what Apache Karaf is:
OSGI is very simple. You can start an OSGI runtime in your application and not even notice it. The shell, the ability to hot deploy from a folder, install from a maven repo, etc..These are all extras that OSGI doesn't have to do and are provided by additional projects like Karaf, GOGO, or Pax
Now on to Apache Karaf:
It is basically an OSGI environment that provides some additional goodies on top of a standard OSGI implementation. Because Apache Karaf is just built on standard OSGI, it can in theory run on any OSGI runtime. This gives you a uniform interface for working with OSGI runtimes.
Some goodies Apache Karaf provides that you won't see in a Apache Felix without some additional work:
Folder based hot deployment
A (IMHO) better default console than gogo. (org.apache.karaf.shell)
Remote SSH access to that console. (org.apache.karaf.shell.ssh)
I mentioned Karaf having it's own way of provisioning bundles and start levels. Apache Karaf also comes with a bunch of pre-defined ones to get started with. I know Apache Felix comes with some as well, through OBR, but Apache Karaf comes much more 'batteries-included'.
Apache Karaf is a swiss army knife to run OSGi "stuff" and non OSGi things bundled as OSGi.
Felix and Aries are default but the frameworks are pluggable, so equinox works as well.
"Unix like" shell accessible from CLI or SSH
Hotdeploy of bundles
May deploy almost anything as a bundle (plain .jar, spring xml, blueprint, etc)
WAR files
Deploy from maven - including autowrapping non OSGi bundles.
Web Console
Karaf features - ".kar files" that includes bundles and custom karaf features/commands.
A side note is that Karaf is part of ServiceMix (the initial name was ServiceMix kernel), so it has very tight integration with Apache Camel, Apache CXF and Apache ActiveMQ. Turning Karaf into an ESB with those features is trivial.