I am installing JBoss but I don't understand which version to choose. Should I download JBoss 7.x or the JBoss EAP 6.x? What is the difference?
TLDR; out of those options, pick the latest EAP 6.x Alpha.
Note that Wildfly was released 11.2.2014, which is now the preferred version.
The long story:
Previously, JBoss community edition was free for all, and EAP was the enterprise version for which you'd need a commercial subscription for. In return you'd get commercial support for it as well. However, with later version things changed a bit and this is no longer exactly so.
JBoss AS7 is JBoss version 7.1. For JBoss 7.2 version, they released it as JBoss EAP 6.1 Alpha, so that is the updated community version. Alpha is free to use in any way or form. They have also released JBoss EAP 6.1 (the commercial version), for which you need a subscription for. If you take a look at the download page, you see the versions together with the release dates:
Version Release date
6.1.0 Final 2013-05-20
6.1.0 Beta 2013-04-19
6.1.0 Alpha 2013-03-07
7.1.1.Final 2012-03-09
6.1.0 Final and 6.1.0 Beta binaries are for commercial purposes and require a paid subscription to be run in production environments.
There are some considerable improvements in 7.2, and 7.1 is very old already, so for 7.x community version you want JBoss EAP 6.1 Alpha or later 6.x Alpha.
Sources:
Update 20.7.2015: since writing the previous answer, JBoss EAP 6.1 Alpha has been superseded by newer JBoss 6.x EAP Alphas in the AS7 product line, latest being JBoss EAP 6.4 Alpha (see "view older downloads"). Statements about 6.1 Alpha release cover all other Alpha releases in EAP 6.X product line as well, so no reason to go for 6.1 Alpha anymore, but to latest 6.x Alpha.
Also, previous download page has been changed to not include EAP releases anymore. They can be found only at EAP download page, and it doesn't even contain 6.1 Alpha anymore.
And finally: JBoss 7.x line is nowadays superseded by Wildfly and EAP 6.x line by EAP 7.