$ perl -MYAPE::Regex::Explain -e 'print YAPE::Regex::Explain->new(qr/^
+(.+)_[^_]+$/)->explain' 

The regular expression:

(?-imsx:^(.+)_[^_]+$)

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):
----------------------------------------------------------------------
  ^                        the beginning of the string
----------------------------------------------------------------------
  (                        group and capture to \1:
----------------------------------------------------------------------
    .+                       any character except \n (1 or more times
                             (matching the most amount possible))
----------------------------------------------------------------------
  )                        end of \1
----------------------------------------------------------------------
  _                        '_'
----------------------------------------------------------------------
  [^_]+                    any character except: '_' (1 or more times
                           (matching the most amount possible))
----------------------------------------------------------------------
  $                        before an optional \n, and the end of the
                           string
----------------------------------------------------------------------
)                        end of grouping
----------------------------------------------------------------------
[download]

-enlil

First, let me say that this is one of the best explanations of a regex that I have seen.

Now, can anyone give a good example of 1 or 2 lines that would match and 1 or 2 that would NOT match?

Allow me to reword and provide a possible match example:
Match a line with at least one underscore (if more than one, the following applies ONLY to the LAST underscore) that is both preceeded AND followed by at least one character.

Match: I_love
Match: I_love_perl
Match: I_love_perl_!
No Match: I_love_perl!_

Am I right?

What a neat module. I was un-aware of this until seeing this post!

It matches and captures at least one character and as any as it can up to (but not including) the last underscore in a string. It fails if there is not an underscore in the string and if there is not at least one character before and after the underscore.

That code stores in $1 a copy of $b with everything after the last "_" removed, including the "_" itself.

There are exceptions.
$1 is left unchanged if $b contains no "_".
$1 is left unchanged if $b contains nothing before the last "_".
$1 is left unchanged if $b contains nothing after the last "_".

When executed in list context, it returns $1 on match or an empty list on failure to match.

$b='This_is_Sample_Text';

if ($b=~/^(.+)_[^_]+$/)
{
    print "\n$1"; ##  This_is_Sample
}
[download]

. = Match any character (except newline)

+ = Match 1 or more times

.+_ = Match upto last underscore(_)

[^_]+$ = Notmatch underscore(_) upto end of the string

Erudil has created a single page Perl Regular Expression Quick Reference (PDF), which may prove useful to you if you have similar questions in the future.