$ (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.

Output.

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

Code.

Output.

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.

! (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.

$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