m#/([^/]*)$#;
[download]

Tip #9 from the Basic debugging checklist: YAPE::Regex::Explain. But, due to some limitations, I had to give the tool a little help:

The regular expression:

matches as follows:
  
NODE                     EXPLANATION
----------------------------------------------------------------------
(?-imsx:                 group, but do not capture (case-sensitive)
                         (with ^ and $ matching normally) (with . not
                         matching \n) (matching whitespace and #
                         normally):
----------------------------------------------------------------------
  /                        '/'
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    [^/]*                    any character except: '/' (0 or more
                             times (matching the most amount
                             possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
[download]

Customarily, // are used as delimiters, but since it wants to match /, alternate delimiters are used (##) to avoid excessive escaping.

See also:

• Read Writeup Formatting Tips and update your post to add "code" tags around your code.
• perlop
• File::Basename