I know this sounds very simplistic, but you'd be surprised how many people simply have no clue how to do this:
You want to update Column1 in TableA to match Column1 in TableB as long as Column2 in TableA matches Column2 in TableB:
Here's how the query should look:
UPDATE TableA
SET Column1 = B.Column1
FROM TableA A
INNER JOIN TableB B
ON A.Column2 = B.Column2