In JBossAS, you use a *-ds.xml file to set up a data source, which is then used to maintain database connections. See the docs for more details.
You can use the JMX Console to monitor a connection. In a browser enter http://localhost:8080/jmx-console. You will be shown a long list on mbeans. Scroll down to a bean named jboss.jca:name=DSNAME,service=ManagedConnectionPool, where DSNAME is the name of your data source (this name comes from the *-ds.xml file). Click on that link and you will be shown the current connection statistics.
Finally, if there is an issue with a connection, an exception stack trace usually shows up in the logs.