The other day I decided to reinstall everything on my server. Not knowing there was an alternative (mysqldump), I started by backing up the entire databases directory (/var/lib/mysql). Now I have that databases directory on a CD, but when I copy all of the files from a database to a new database with root, only the root mysql user has write permissions. How can I give the correct owner read and write access to their database?

You can create the required users and give necessary permissions to the corresponding databases using the GRANT command.
http://dev.mysql.com/doc/refman/5.0/en/grant.html gives the usage of the command.

Thanks a lot.

You are welcome

That did not solve the problem. I still get an error saying that the tables are read-only.

Sounds like the ownerships of the files in /var/lib/mysql are incorrect; you probably need to chown them so that the mysql user (or whichever user runs mysqld) has read/write permissions to them.

Check to see what they were on the originating computer and use that as a guide; also check the names of the user which runs mysqld on both sides.

Or if you have the time/bandwidth to re-copy them, use tar instead of cp:

# cd /var/lib/mysql
# tar cfpz mysql.tgz

...then extract that on the other side with

# tar xvfpz mysql.tgz

(It might be a good idea to back up your original copy first.)