You are up to date with 10.4.11 Tiger. 11 was the last update.
In System Preferences > Software Update you can check on Installed Updates to see what you have. I expect you will have them all, as the previous owner probably did them.
I am not sure of the last security update but it may be
Security Update 2007-009 (OS X v10.4.11 PPC)
Note your eMac is a PPC type processor. The other one is Intel. They both can run 10.4 Tiger so that is why the two types are listed. Yours is a PowerPC Mac.
Safari for 10.4 is no longer maintained by Apple so you need to use something else.
A good web browser is TenFourFox that is still maintained by its developers.
It will do 10.5.8 respectably then.