You are reading a line from the terminal, not just a single character. You need chomp to remove the line ending character(s). Also, get in the habit of indenting your code one level when looping.

print "please enter one character: ";
$char = <>;
chomp $char;
while ($char ne ".") 
{
   if ($char eq "a") {
    $count++;
   }
   print "please enter one character: ";
   $char = <>;
   chomp $char;
}

print "The number of a is $count\n";
print "done";
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