Upgrading and Migrating Joomla the easy way!
Even though Joomla is a popular Content Management System (CMS) used by millions of websites, upgrading from one one version of Joomla to another is still not systematized by the Joomla or Open Source Matters! (the company behind this popular CMS.)
Upgrading Joomla refers to elevating in Joomla’s minor version – for example from 1.5 to 1.5.x while Migrating Joomla refers to elevating from Joomla v1.5 to v1.6 (so called major version). The next major release will be Joomla version 2.
Upgrading and migrating Joomla is not that easy as is with WordPress (world’s most popular blogging client now used as a powerful CMS). The Joomla Wiki Page (documentation) has a long and complicated process, you can read here Upgrading 1.5 from an existing 1.5x version. However, here is the easiest way to upgrade Joomla 1.5 existing version to Joomla 1.5.x using Update Manager for Joomla and migrating from Joomla 1.5 (latest 1.5.x) version to Joomla 1.6 using jUpgrade – both free components for Joomla! In two series, you can update from your 1.5 Joomla to 1.6 version.
How to upgrade Joomla 1.5 to an existing Joomla 1.5.x version?
!!! Please, backup your Joomla website before working !!!
Step 1. Download and install a free Joomla component called Update Manager for Joomla! into your Joomla backend.
Step 2. Open Update Manager from the Components menu. Joomla! Update Manager will guide you through the upgrading Joomla! installation in easy way. No need to panic right now, simply continue to download the update files >>>
Step 3. Optionally there are configuration parameters for Update Manager which can be accessed from the right top corner, unless you are using proxy you do not need to change the settings. (Most users can skip this process)
Step 4. Joomla! Update Manager – Step 1 will show your currently running Joomla version (1.5.15 as in screenshot) and gives you the option to download either the Patch Package or the Full Package. Since we need to update our existing Joomla installation to the latest release (here 1.5.15 to 1.5.22), we select the Patch Package.
Step 5. Continue on next Step. Proceed with the install.
Step 6. You will be congratulated with the successfully upgraded Joomla! installation screen. That’s it. Now your Joomla installation is upto date on that version (Joomla 1.5.22 is the latest version released for Joomla 1.5)
This process is expected to be applied to upgrading from 1.6 to 1.6.x versions as well.
Note: If you are wondering how to check the current Joomla! version of your website then, go to System Info from Help menu (process shown in screenshots below)
How to migrate from Joomla 1.5 to Joomla 1.6 version?
Getting Joomla 1.6 from Joomla 1.5 is the migration, but there is no core upgrade path (which is expected to be available for Joomla 2). According to Joomla Wiki page, In planned future releases of Joomla (which will be released every 6 months), such as Joomla 1.7, 1.8, etc, the changes from version to version will be more incremental and a core upgrade path is planned. 
!!! Please, backup your Joomla website before working & read jUpgrade requirements !!!
Step 1. Install jUpgrade and then enable Mootools upgrade plugin
Step 2. Open jUpgrade from Component Menu.
Step 3. Start upgrade, let jUpgrade work for you; it will download Joomla! 1.6 and then decompress the package. If available it will upgrade the 3rd party extensions as well.
Step 4. What next, it is all done. Just check your site and admin section. Ensure that as of current version, jUpgrade doesn’t migrate templates from Joomla 1.5 to 1.6. You should work to make Joomla 1.5 templates in Joomla 1.6.
Please, read the detail requirements and documentations on Migrating from Joomla 1.5 to Joomla 1.6 from official Joomla Wiki here. The video tutorial below will help you a lot.