Both davorg and MZSanford have pointed out above that you're using symbolic references and hinted why that's a bad idea ;-) Just to complete the picture, here's what your code would look like using true or "hard" references.

use strict 'refs';
# Catches use of symbolic references,
# though plain "use strict" would be even better

# Quotes are not necessary on the LHS of =>
%simpsons = (
    father => "homer",
    mother => "marge",
    son => "bart",
    daughter => "lisa",
    );

# Here's the code that uses symbolic refs:
#$string = "simpsons";
#@array = ("simpsons");
#%hash = ("family" => "simpsons");
#print("$simpsons{father}\n"); #case1 prints "homer"
#print("$$string{mother}\n"); #case2 prints "marge"
#print("$$array[0]{son}\n"); #case3 err
#print("$$hash{family}{daughter}\n"); #case4

# Here are the hard refs.
$string = \%simpsons;
@array =  (\%simpsons);
%hash  = (family => \%simpsons);

# Not necessary to quote variables to print
print $simpsons{father}, "\n";          #case1 prints "homer"
print $$string{mother}, "\n";           #case2 prints "marge"
print $string->{mother}, "\n";          #also prints "marge"
print ${$array[0]}{son}, "\n";          #case3 prints "bart"
print $array[0]->{son}, "\n";           #also prints "bart"
print ${$hash{family}}{daughter}, "\n"; #"lisa"
print $hash{family}->{daughter}, "\n";  #also "lisa"
print $hash{family}{daughter}, "\n";    #also "lisa"
[download]

For more information on the backslash operator for taking references and the syntax for getting at its contents, take a look at the doco on references.

HTH