I had trouble with local when I was first getting started too. I actually found it to be a lot simpler then I had made it out to be though. Here are some definitions:

# Define a global variable, the compatible way
use vars qw($foo);

# Define a global variable, the Perl 5.6.x way
our $foo;

# Define a "lexical" variable
my $foo;
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Local comes in to play when you have a global/package variable defined, and you want to temporarily use that variable for something else. It often comes up with the special variables, such as $/, which is the variable which defines the input seperator. By default, it is the newline. But what if you would like to suck in an entire file, and not do it line by line? You could do this, not using local:

undef $/;             # Enables whole file mode
open (FH, $filename); # Opens the file 
while(<FH>) {         # Contains the whole file
  blah
}
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That works. However, it's not considered "safe". What is considered more safe would be to use local, which would limit the scode of $/ when we undefine it like so:

{
   undef local $/;
   open (FH, $filename);
   while(<FH>) {
      blah
   }
}
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When the last closing bracket is reached, the original value of $/ (newline by default) is restored, as the "local" definition goes out of scope.

Hope that helps!
-Eric

Update: Whoops! As Hofmator pointed out, I appear to have forgotten to actually use the local statement :-) It's now fixed. Thanks Hofmator.