There shouldn't be a FROM clause in the UPDATE statement, and the SET clause should follow the full set of table references:
UPDATE  table1 t1 
JOIN    table2 t2 ON t1.ID = t2.ID
SET     t1.col1 = 1
WHERE   t1.Name = 'Test' AND t2.Age = 25;
Test case:
CREATE TABLE table1 (id int, col1 int, name varchar(20));
CREATE TABLE table2 (id int, age int);
INSERT INTO table1 VALUES (1, 0, 'Test');
INSERT INTO table1 VALUES (2, 0, 'Test');
INSERT INTO table1 VALUES (3, 0, 'No Test');
INSERT INTO table2 VALUES (1, 20);
INSERT INTO table2 VALUES (2, 25);
INSERT INTO table2 VALUES (3, 25);
Result:
SELECT * FROM table1;
+------+------+---------+
| id   | col1 | name    |
+------+------+---------+
|    1 |    0 | Test    |
|    2 |    1 | Test    |
|    3 |    0 | No Test |
+------+------+---------+
3 rows in set (0.00 sec)