MySQL Insert Into from one Database in another

Your query should go like this:

INSERT INTO newDatabase.table1 (Column1, Column2) 
SELECT column1, column2 FROM oldDatabase.table1;

UPDATE

Since this answer is getting more attention than I even anticipated, I should expand on this answer. First of all, it might not be obvious from the answer itself, but the columns do not need to have the same name. So, following will work too (assuming that the columns exist in their respective tables):

INSERT INTO newDatabase.table1 (Column1, Column2) 
SELECT SomeOtherColumn, MoreColumns FROM oldDatabase.table1;

Furthermore, they don't even need to be real columns in the table. One of the examples for transforming data that I use quite often is:

INSERT INTO newDatabase.users (name, city, email, username, added_by) 
SELECT CONCAT(first_name, ' ', last_name), 'Asgard', CONCAT(first_name,'@gmail.com'), CONCAT(first_name,last_name), 'Damir' FROM oldDatabase.old_users;

So, as it might be more obvious now, the rule is, as long as the SELECT query returns same number of columns that INSERT query needs, it can be used in place of VALUES.

INSERT INTO db1.table SELECT * FROM db2.table;

If you want to copy data to same tables of different db.

You said "The tables and columns got different names", but you still used the same names. Try this:

INSERT INTO newDatabase.newtable1 (newColumn1, newColumn2) 
SELECT oldcolumn1, oldcolumn2 FROM oldDatabase.oldtable1;