How do we know that a directory is apache writable?

Apache, is a program running in the background. Apache is originally started by user root (also called root-process). This root-process launches several child processes which handle the client requests. For security reasons, the child processes are not run by user root but as a user with minimal privileges. Usually this user is named apache or www-data.

To find out what user this is for apache v1:

ps -ef | grep httpd | grep -v grep

or for apache v2:

ps -ef | grep apache | grep -v grep

Result for apache 2 will be something like this:

root      5001     1  0 07:21 ?        00:00:00 /usr/sbin/apache2 -k start
www-data  5021  5001  0 07:21 ?        00:00:00 /usr/sbin/apache2 -k start
www-data  5022  5001  0 07:21 ?        00:00:00 /usr/sbin/apache2 -k start
www-data  5023  5001  0 07:21 ?        00:00:00 /usr/sbin/apache2 -k start

In this case the user/group is www-data

So, in order to make a directory writable by the webserver we have to set the directory’s owner or group to Apache’s owner or group and enable the write permission for it. Usually, we set the directory to belong to the Apache group (apache or `www-data or whatever user is used to launch the child processes) and enable the write permission for the group.

chgrp www-data /path/to/mydir
chmod g+w /path/to/mydir

(www-data is the name you found with the ps command above).

Regarding:

2) I've also been told to make the app/runtime directory web-writable. Is this the same as apache writable ?

Yes, this is a directory you need to set writable to the group Apache expects. Probably this will be somewhere in /var/www/ or it is set as a virtual host in /etc/apache2/sites-enabled/ and/or /etc/apache2/sites-available