I am not aware that there is a diagram on how playsms works. Probably Anton has one and can help you in the matter but in general the process goes this way (keep in mind that the actual process is way more complicated but as an overview this is how it works).
Remember that playsms does not actually sends or receive the messages you use a gateway. Now that gateway can be a commercial provider or you can set up your own using sms server tools, kannel or gammu. If you will use a commercial provider make sure that is supported by Playsms by checking Settings => Manage gateway and SMSC.
If you are going to set up your own gateway you can do it with a gsm modem or just an old phone. that is compatible with any of those software that I mention.
I personally use gammu with very good results and it seems that is the one that is less complicated. I am not sure about sms server tools or kannel but you are welcome to give them a try.
So this is how it works with gammu. Your modem or old phone is connected to your server via a serial cable or usb cable. Depending on the phone you can even connect with it even using bluetooth. So when it receives a message gammu sense it a move the message from the phone to a folder in you computer. From there Playsms daemon pick it up and process it. You can choose what to do with the message. Usually if the message comes with a keyword playsms will do something (like running a external command, deliver it to a inbox or group etc.)
When you send, playsms process the message either from the webpage directly or from webservices and creates a file with a specific name and format and place it in a directory in your server. From there gammu pick it up and send it to your phone with the instructions to send it to a specific number.
In general this how it works. You can see a list of compatible phones and modems (with gammu) here: http://wammu.eu/phones/
Also you can see the documentation for the smsd daemon here http://wammu.eu/docs/manual/smsd/index.html#smsd
The compilation instructions are here http://wammu.eu/docs/manual/project/install.html
Here is guide on how to install on ubuntu: http://xaintech.blogspot.com/2010/05/build-your-own-sms-gateway-with-ubuntu.html
Here is another: https://pakeklinux.wordpress.com/2011/07/09/how-to-install-and-configure-latest-gammu-1-29-in-ubuntu-10-04/
You can check the documentation for playsms here: http://book.playsms.org/
Once you have your gammu-smsd working to set up playsms and start sending messages is very easy
Remember that you have to have a LAMP up a running before you can start with playsms. Here some documentation about that in ubuntu https://help.ubuntu.com/community/ApacheMySQLPHP
I hope this helps. Please let me know if you have more questions.