Move the WHERE clause to after the Joins (which are actually part of the From clause), and I suspect you want a Left join (as you implied the other table may not have matching rows)
Code:
SELECT COLUMN_NAME, `table2`.fieldName
FROM INFORMATION_SCHEMA.COLUMNS 
LEFT JOIN `common_field_names_table`
WHERE TABLE_SCHEMA='myDB' AND TABLE_NAME='table1'
ON `INFORMATION_SCHEMA`.`COLUMN_NAME` = `table2`.fieldName