You can install the software on your local server. You then will have to maintain it yourself.

The main difference between the version proposed by odoo is that it only allows you to install the certified modules. If you want to choose the modules you want to install, you will have to install it on your own (local or remote) server. You need to know that some of the non certified modules can bring the whole system to stop from restarting, so that there is some risk with it, that is not relevant if you only do some testing and want to learn the technical part of Odoo.

If you want to try the system from a functional point of view, you are better off with creating a database with Odoo and install the modules you want to use. This online version allows you to go very deep in the technical side, like adding fields, changing the layout, changing the workflow, playing with the authorisation, etc. So, it is a very good starting point in the Odoo world.

Odoo is like an OS called Odoo server, you can select from apps (which are Modules) available and install them, just like you install apps to you computer and Smart phones. Running online means you need to create an account with odoo and "install" the modules you want to use online (hosted by Odoo).

I think "installing it" should mean you download the Odoo installer from: https://www.odoo.com/page/download and run the installer on your computer. After which you will also install the apps that you want say CRM, Accounting etc. Installing it means also using your local server called localhost. This doesnot require internet to use but will require internet to install apps. This should give you a good learning environment. You can use the online for your production environment. Cheers!

It's possible to install on your local server indeed. I agree with Fabien that small companies typically do not have the capital nor bandwith to maintain the hardware, meaning SAAS solutions are preferred. However I do see some room for improvement still on the SAAS version when comparing e.g. to functionality provided by Salesforce or SAP Business ONE. Some challenges I ran into thus far when testing the SAAS version of ODOO:

1) Only certified modules can be installed. The set of certified modules is quite extensive indeed but I do miss some functionality like (cron) batch scheduling e.g. which is available in the library but can only be added when installing ODOO on your own server. I also miss the opportunity to easily add functionality yourself like e.g. Salesforce is offering with Visualforce and Apex.

2) Integration with 3rd party software and ODOO specific apps. I installed the ODOO smartphone app as I like the idea to manage (purchase) orders on the go (or when talking to customers/vendors) but it doesn't seem possible to login to the .odoo.com accounts. Similarly I tried to integrate new leads/opportunities from Salesforce.com and weborders from Wordpress Woocommerce into my .odoo.com account using the Zapier integration service but also this doesn't seem possible (yet) for odoo.com accounts (it is possible to use the Zapier integration service when installing ODOO on your own server).

3) As I'm only test-driving the software right now I tried to stay below the 3 user limit to avoid having to pay cost. When trying to create a new portal user (which should not be charged) I accidently created a new user when I found out that the additional fee that comes as of the 3rd user has to be paid by installed app. As I installed quite some apps the monthly cost all of a sudden increased from 0 to 240 EUR per month. In my opinion a better business model would be to do charge a fee as of the 3rd user but not per app as this might prevent clients to test drive new functionality.

4) Allowing external customers/vendors access to the online portal/webshop is quite cumbersome. Portal users are forced into creating an ODOO account first which raises many questions. Also the promotion of ODOO on the webshop is not appreciated by all clients, meaning many clients will prefer to use alternate ecommerce solutions instead and have them integrate the sales orders with ODOO.

I do think ODOO is heading into the right direction though and what I've seen from Fabien and his team thus far I'm sure they will try to address the concerns as raised by me above.

What is the difference between running it online and installing it?

Sorry

But if I would like  to install in my Local Server it is possible yet ???