Your question isn't really related to Perl; it's more of a Unix/Linux question.

In the examples you gave:

1. /dir/dir/
2. /dir/dir
3. dir/dir/
4. dir/dir
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Numbers 1 and 2 are absolute paths, meaning that they start at (ie. "are anchored at") the root directory "/".

Numbers 3 and 4 and relative paths, meaning that there location depends on which directory you are in when you refer to them.  For example, if you are in the absolute directory "/var/www/html", and you refer to "css/styles", you are accessing the absolute directory "/var/www/html/css/styles".

But if you started in the directory "/var/www/cgi", then referring to "css/styles" would mean an absolute path of "/var/www/cgi/css/styles" instead.

Other than that, "/dir/dir/" and "/dir/dir" are nearly always identical.  There are a few places where they behave in subtly different ways (eg. if "/dir/dir" is a symbolic link), but they are not anything you need to worry about at this stage in your learning curve.