The compressed folder contains both plain-text and gzipped compressed versions of the floatbox files. The plain-text versions are compressed using the YUI compressor. The gzipped files are zipped from the YUI compressed files.
To use these compressed files on your site (probably a good idea), point your floatbox include lines to the "compressed" folder.
For example, if floatbox is installed in /includes/floatbox/, your pages would use the uncompressed floatbox files with the header lines of:
<script type="text/javascript" src="/includes/floatbox/floatbox.js"></script>
<link rel="stylesheet" type="text/css" href="/includes/floatbox/floatbox.css" />
And to use the compressed files, your header lines would be:
<script type="text/javascript" src="/includes/floatbox/compressed/floatbox.js"></script>
<link rel="stylesheet" type="text/css" href="/includes/floatbox/compressed/floatbox.css" />
WARNING: Do not copy the compressed files down into the main uncompressed floatbox folder. If you do, the css files will not be able to find the graphics folder (without modification).
To get the best compression, you want to deliver the gzipped files. To do this on an Apache server, you could try renaming the "htaccess" file in the compressed folder to ".htaccess". WARNING: If you are using directory aliases, you will need to edit the RewriteBase path in the .htaccess file. (This is one reason why .htaccess is not enabled in the download by default.) Using mod_rewrite and .htaccess files is very much a mysterious black art. What works on one server setup may not work on another one. You may have to merge the gzip stuff here in with your root folder's .htaccess file. You may have to do something else. If you don't know what I'm talking about, don't bother trying. You will find only grief.
If you use the gzipped file version of floatbox.js, you are stuck with the default options (unless you build your own .gz file). In this case, use page and anchor options to set your preferences. (See the docs).
Lastly, if this is confusing or you're having trouble getting it all too work, just use the normal uncompressed files in the normal uncompressed way. They work fine and the compressed ones are really for the fanatics (you know who you are).