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$ (dollar sign)
Declare/use a variable
Example.

$abc = "123"

$_ (dollar underscore)

'THIS' token. Typically refers to the item inside a foreach loop.
Task: Print all items in a collection.
Solution.

... | foreach { Write-Host $_ }

$$ (double dollar, two dollars) 
Last entered token. Does NOT refer to a whole command.
Task: Say hello world twice.
Solution.

Write-Host "Hello, world!"
$$

$? (dollar sign + question mark) 
Many google searches were looking for this information, so I experimentally found what it does.

Returns True or False value indicating whether previous command ended with an error. For some reason it does not catch all errors, but most of the time it works. 
Task 1: See if a powershell cmdlet exists in the system.
Code.

SomeCmdLet #does not exists
$?

$?

Output.

The term 'SomeCmdLet' is not recognized as the name of a cmdlet, function, script file, or operable program. Check the

spelling of the name, or if a path was included, verify that the path is correct and try again.

At line:1 char:15

+     SomeCmdLet <<<<  #does not exists

    + CategoryInfo          : ObjectNotFound: (SomeCmdLet:String) [], CommandNotFoundException

    + FullyQualifiedErrorId : CommandNotFoundException

False    #error occured - previous cmdlet (SomeCmdLet) was not found

True     #no errors returned by the previous command ($?)

Task 2: See if a WMI class exists in the system

Code.

gwmi win32_processo -ErrorAction SilentlyContinue   #intentional error, win32_processor is the right one

$?

$?

Output.

| (pipeline)
Catch output of the command and pass it to another command.
Task: Get list of processes and select top 3 items.
Solution.

Get-Process | Select-Object -first 3

% (percentage) 
1. Shortcut to foreach.
Task: Print all items in a collection.
Solution.

... | % { Write-Host $_ }

2. Same as Mod in VB.
Example:

5 % 2

.. (double dot) 
Specify a range.
Task: Print numbers 1 through 5 without a foreach loop.
Code.

1..5

Output.

1
2
3
4
5

:: (double-colon) 
Thanks to Darrell for asking about this one.

Reference static member of a class.

Task: Compare two strings.
Code.

[string]::Equals("a", "b")

Output.

False

! (exclamation mark)
Thanks to Leo for asking about this one. 
Shortcut to -not.
Code.

$a = $null;
if(!$a) { Write-Host '$a is null' }

Output.

$a is null

? (question mark)
Output all items that conform with condition (shortcut to where). Shortcut to the following: 

foreach { if (...) {return ... } }

Task: Print all odd numbers between 1 and 5 (inclusive):
Code.

1..5 | ? { $_ % 2 }

Output.

1
3
5

` (backtick)
1. Continue command on the next line.
Code.

Write-Host `
"Hello, world!"

Output.

Hello, world!

2. Include a special symbol into a string. Available options:
`0 - Null. My preference is using $null instead.
`a - Alert. Yes, it does make sound, and you can use multiple for multiple beeps.
`b - Backspace
`f - form feed - only affects printed documents.
`n - New line
`r - Carriage return
`t - Horizontal tab
`v - Vertical tab - only affects printed documents.
`' - Single quote - I prefer using double quotes when I need to escape a single one, since I don't need any escaping in this case.
`" - Double quote - you can use single quotes, and you don't need this symbol. My preference is use standard escaping instead, so 4 double quotes ("""") means a double quote.
Official article by Microsoft with full description on every token: http://technet.microsoft.com/en-us/library/hh847835.aspx

# (pound sign)
Single line comment.
Example.

#This is a commented line
#This is a second one

<# ... #> (less triangle / lt + pound ... pound + greater triangle / gt)

Block/Multi-line comment.
Example.

<#This is
a commented
block#>

& (ampersand)
Execute string as command.
Example.

& "Get-Process"

@( ) (email-at + round brackets)
Declare arrays.

Note: comma is used as a separator, in contrast to hash table declaration.
Example.

$a = @("One", "Two", "Three")

@{ } (email-at + curly brackets/braces)
Declare hash tables.

Note: semicolon is used as a separator, in contrast to array declaration.

Example.  

$a = @{"1" = "one"; "2" = "two"; "3" = "three"}

@' ... '@ (email-at + single quote ... single quote + email-at)
Here-string without embedded variables.
Code.

@'
$(1+2)
$(3+4)
$(5+6) 
'@

Output.

$(1+2)
$(3+4)
$(5+6)

@" ... "@ (email-at + double quote ... double quote + email-at)
Here-string with embedded variables.
Code.

@"
$(1+2)
$(3+4)
$(5+6)
"@

Output.

3
7
11