I have a textures.jar file on my server (specified with the archive parameter within the applet tag) with a class to use as an anchor and some images. When I run the applet, this is how I get to the images:
Does this mean that every time I use a getResource() on an applet it goes to the server for it?
No. Generally the Java Plugin will cache jar files. When the applet is loaded the Plugin checks whether the file on the server is newer than the one in the cache; if it is, it will replace the existing cached copy, but otherwise the cached copy will be used.
Note that the user can turn off jar file caching in the Plugin control panel, but by default it's turned on (and most users probably never touch those settings).