The -eq (equal to) and -ne (not equal to) operators perform exact (and, by default, case insensitive) comparisons. In the example below, the following returns true:

1 -eq 1 
'string' -eq 'string' 
[char]'a' -eq 'A' 
$true -eq 1 
$false -eq 0 

Similarly, -ne (not equal) will return true for each of the following:

20 -ne 100 
'this' -ne 'that' 
$false -ne 'false' 

The last example compares $false, the Boolean, with a string containing the word false. PowerShell will attempt to convert the word, but as the word is not an empty string, the result will be true.