Comparison Operators

The following operators are all Case-Insensitive by default:

 -eq             Equal
 -ne             Not equal
 -ge             Greater than or equal
 -gt             Greater than
 -lt             Less than
 -le             Less than or equal
 -like           Wildcard comparison
 -notlike        Wildcard comparison
 -match          Regular expression comparison
 -notmatch       Regular expression comparison
 -replace        Replace operator
 -contains       Containment operator
 -notcontains    Containment operator

To perform a Case-Sensitive comparison just prefix any of the above with "c"
for example -ceq for case-sensitive Equals or -creplace for case-sensitive replace.

Similarly prefixing with "i" will explicitly make the operator case insensitive.

Types
 -is     Is of a type
 -isnot  Is not of a type
 -as     As a type, no error if conversion fails

Logical operators
 -and    Logical And
 -or     Logical Or
 -not    logical not
  !      logical not

Bitwise operators 
 -band   Bitwise and
 -bor    Bitwise or
 
Format Operator
 "format_string" -f Object1, Object2,...

The format_string is in the form: {0,-5} {1,-20} {2,-10}
In each set of braces, the first number, before the comma refers to the column.
The second number, after the comma determines the padding (how many characters)
If the second number is negative, it not only pads the element, but aligns it vertically. Optionally the second number can be used for formatting :hh :mm :C :p

Examples

$myVar -is "String"
$myVar -eq 123
$myVar -ceq $myVar2
"abcdef" -like "abc*"
"abcdef" -replace "dEf","xyz"
$myVar1 -is "String" -and $myVar2 -is "Int"
"{2:N}" -f 24.4567
(1 -eq 1) -and -not (2 -gt 2)

$mycmd=ps|select id,ProcessName
foreach ($proc in $mycmd) {"{0,-8}{1,-20}" -f $proc.id, $proc.ProcessName}

“The 50-50-90 rule: Anytime you have a 50-50 chance of getting something right, there's a 90% probability you'll get it wrong” ~ Andy Rooney

Related:

if - Conditionally perform a command
Assignment Operators ( $variable=X, $variable +=Y )
PowerShell Regular Expressions
Format-Table