There's really no relation between those functions, and my is the only one which declares variables. In short,

• my creates a lexically scoped variable. That means it only exists within the current block/file. For example,

  {
     my $var;
  }
  # $var no longer exists.
  [download]

• our is the same as no strict 'vars';, except is only affects the specified (package) var or vars. Like use strict, its effect is bound to the current block/file. For example,

  package main;
  
  use strict;
  
  $main::var = 1;
  #$var = 1;              # Error!
  
  {
     $main::var++;
     #$var++;             # Error!
     our $var;            # Disable "strict vars" for $var.
     $var++;              # Same as "$main::var++;"
  }                       # "strict vars" is back on for $var.
  
  {
     $main::var++;
     #$var++;             # Error!
     no strict 'vars';    # Disable "strict vars" for all vars.
     $var++;              # Same as "$main::var++;"
  }                       # "strict vars" is back on for all vars.
  
  print("$main::var\n");  # Prints 5.
  #print("$var\n");       # Error!
  [download]

• local saves the current value of a package variable, and restores it when the current block is exited, even if it is exited by exception. For example,

  $main::var = 1;
  
  {
     local $main::var;  # Saves current value of $main::var
                        # and sets new value to undef.
     
     $main::var = 2;
  } # Restores saved value of $main::var.
  
  print("$main::var\n");  # Prints 1.
  [download]

  which is the same as

  package main;
  
  our $var;
  $var = 1;
  
  {
     local $var;        # Saves current value of $main::var
                        # and sets new value to undef.
    
     $var = 2;
  } # Restores saved value of $main::var.
  
  print("$var\n");      # Prints 1.
  [download]

Now, you're probably wondering when to use lexical variables and when to use package variables.

Again in short, you want to use lexicals whenever possible. There are a few situations that mandate using pacakge variables:

• Package variables are used for global variables which need to be accessed from other modules. For example, @ISA and @EXPORT are package variables.

• Package variables are also used when dynamic scoping is needed, because of local makes it easy to do so. For example,

  sub recursive {
     our $state;
  
     ...
  
     if (...) {
        local $state = ~~~;
        recursive();
     }
  
     ...
  }
  [download]

• Package variables are needed for (?{ ... }) and similar constructs in regexps.

• Aliases are created using package variables (except when using some modules). For example,

  my $var = 1;
  our $alias;
  *alias = \$var;
  $alias = 2;
  print("$var\n");   # 2
  [download]

The variables in perlvar are package variables, so you can use local on them.

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