How to assign a select result to a variable?

I just had the same problem and...

declare @userId uniqueidentifier
set @userId = (select top 1 UserId from aspnet_Users)

or even shorter:

declare @userId uniqueidentifier
SELECT TOP 1 @userId = UserId FROM aspnet_Users

DECLARE @tmp_key int
DECLARE @get_invckey cursor 

SET @get_invckey = CURSOR FOR 
    SELECT invckey FROM tarinvoice WHERE confirmtocntctkey IS NULL AND tranno LIKE '%115876'

OPEN @get_invckey 

FETCH NEXT FROM @get_invckey INTO @tmp_key

DECLARE @PrimaryContactKey int --or whatever datatype it is

WHILE (@@FETCH_STATUS = 0) 
BEGIN 
    SELECT @PrimaryContactKey=c.PrimaryCntctKey
    FROM tarcustomer c, tarinvoice i
    WHERE i.custkey = c.custkey AND i.invckey = @tmp_key

    UPDATE tarinvoice SET confirmtocntctkey = @PrimaryContactKey WHERE invckey = @tmp_key
    FETCH NEXT FROM @get_invckey INTO @tmp_key
END 

CLOSE @get_invckey
DEALLOCATE @get_invckey

EDIT:
This question has gotten a lot more traction than I would have anticipated. Do note that I'm not advocating the use of the cursor in my answer, but rather showing how to assign the value based on the question.

Try This

SELECT @PrimaryContactKey = c.PrimaryCntctKey
FROM tarcustomer c, tarinvoice i
WHERE i.custkey = c.custkey 
    AND i.invckey = @tmp_key

UPDATE tarinvoice SET confirmtocntctkey = @PrimaryContactKey 
WHERE invckey = @tmp_key
FETCH NEXT FROM @get_invckey INTO @tmp_key

You would declare this variable outside of your loop as just a standard TSQL variable.

I should also note that this is how you would do it for any type of select into a variable, not just when dealing with cursors.