You don't say what result you're expecting.

However, I'd guess that you are expecting @A_counter to be redefined for each iteration of the outer loop. SQL variables aren't scoped like that, so you need to reset @A_counter to zero each time.
You may find this does what you want:

declare @B_counter int
declare @A_counter int
set @A_counter = 0
set @B_counter = 0
WHILE @B_counter <= 10 begin
    
    WHILE  @A_counter <= 10 begin
        
        SET @A_counter = @A_counter + 1
        print 'inner loop'
    end 

    SET @B_counter = @B_counter + 1
    SET @A_counter = 0
    print 'outer loop'
end
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On the other hand, you don't say what result you're after, or what you're getting, so I may be way off.