Query to select max value on join

Your existing query is close to something that you could use but you can get the result easily by making a few changes. By altering your query to use the APPLY operator and implementing CROSS APPLY. This will return the row that meets your requirements. Here's a version that you could use:

SELECT 
  u.Username, 
  u.UserType,
  u.Points,
  lv.Level
FROM Users u
CROSS APPLY
(
  SELECT TOP 1 Level
  FROM Levels l
  WHERE u.UserType = l.UserType
     and l.MinPoints < u.Points
  ORDER BY l.MinPoints desc
) lv;

Here's a SQL Fiddle with a demo. This produces a result:

| Username | UserType | Points |  Level |
|----------|----------|--------|--------|
|     John |        A |    250 | Silver |
|     Mary |        A |    150 | Bronze |
|     Anna |        B |    600 | Bronze |

The following solution uses a common table expression that scans the Levels table once. In this scan, the "next" points level is found using the LEAD() window function, so you have MinPoints (from the row) and MaxPoints (the next MinPoints for the current UserType).

After that, you can simply join the common table expression, lvls, on UserType and the MinPoints/MaxPoints range, like so:

WITH lvls AS (
    SELECT UserType, MinPoints, [Level],
           LEAD(MinPoints, 1, 99999) OVER (
               PARTITION BY UserType
               ORDER BY MinPoints) AS MaxPoints
    FROM Levels)

SELECT U.*, L.[Level]
FROM Users AS U
INNER JOIN lvls AS L ON
    U.UserType=L.UserType AND
    L.MinPoints<=U.Points AND
    L.MaxPoints> U.Points;

The advantage of using the window function is that you eliminate all sorts of recursive solutions and improve performance dramatically. For best performance, you would use the following index on the Levels table:

CREATE UNIQUE INDEX ... ON Levels (UserType, MinPoints) INCLUDE ([Level]);

Why not do it using only the rudimentary operations, INNER JOIN, GROUP BY, and MAX:

SELECT   U1.*,
         L1.Level

FROM     Users AS U1

         INNER JOIN
         (
          SELECT   U2.Username,
                   MAX(L2.MinPoints) AS QualifyingMinPoints
          FROM     Users AS U2
                   INNER JOIN
                   Levels AS L2
                   ON U2.UserType = L2.UserType
          WHERE    L2.MinPoints <= U2.Points
          GROUP BY U2.Username
         ) AS Q
         ON U1.Username = Q.Username

         INNER JOIN
         Levels AS L1
         ON Q.QualifyingMinPoints = L1.MinPoints
            AND U1.UserType = L1.UserType
;