If you want a concise definition, you should check out the manual pages. You can find them in the Library, if you don't already have them.
That said, here is a brief in plain english.

my() creates a lexical variable that is local to the block it's in. If you declare my $foo;and call function my_function(), my_function() will not have access to $foo.

local() on the other hand, creates a variable that is local to the current block, and blocks it calls. so my_function() will be able to get at $foo, if you declare it local $foo;

There is more to it than this, but it's a start.

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