How can I launch SQL Developer?
You can download the latest SQL Developer from Oracle and get the specific instructions here on what is required to do the installation.

Once SQL Developer starts, the next obvious step is to connect to a database and this will depend on how Oracle is configured. While I believe this is a separate issue, for sake of completeness I'll add that by default SQL Developer will use xe for the SID in the connection wizard, which is for express edition, and that may not be sufficient to make a connection. Another way is to try orcl instead of xe since that is default for "regular" Oracle.

It is possible this has been customized depending on the installation and if so you will have to find that information from the person/group that installed Oracle if you do not already know it.

If still having problems trying to connect, look at this link to see if any of the information there is helpful: Oracle 11g listener fails with ORA-12514 and ORA-12505 errors