Installing Joomla 1.6 is not as difficult as some people may think, but the trick is to know the "best way" to install Joomla and be organized with a step-by-step guide. Installing Joomla is going to be very similar to version 1.5, unless you are new to this cms, then you still follow the method I use for myself when creating sites. Please note that this installation process will depend on your server and to ensure Joomla 1.6 will be compatible with your website host provider.
Simple enough...download the most recent version of Joomla from www.joomla.org and then unzip the file to your desktop. Open your FTP program and upload the files and directories - as they are - to the location you want to install Joomla. This will take several minutes, depending on your internet connection, but on average, it should take about 8 minutes because of the amount of files and sub-folder structure the package has (see the screenshot below).

NOTE: Some hosts provides a service which lets you auto install scripts, normally this is called Fantastico, but I recommend installing Joomla yourself to ensure it's done correctly. Some people have found problems when doing a host auto installation.
Next step is to create your MySQL database because without one, your website won't function. You will want to make sure your host has at least the following:
When making your database, please use a unique name and password for it. For the password, I RECOMMEND at least 12 characters with symbols included, like !@#$%^&*() etc...although, make sure your host provider allows symbols because this makes your password a lot stronger. If you find you cannot think of a good password, try http://www.pctools.com/guides/password/ which is what I use.
Setting up Joomla is fairly easy and does not take very long, in fact, it takes me less than 2 minutes! First thing you will want to do is open your favourite browser and go to the url location where you uploaded your Joomla files. For example, it might be this:
http://www.yourwebsite.com (or) http://www.yourwebsite.com/joomla/
From this point on, you will follow the online instructions for installing Joomla 1.6 through each step of its process.
Choose Language - This is where you will select the language choice.

Pre-Installation Check - This is where the installer will review your system setup. If you see any red warnings, you may want to look into them, but in some cases, your host may need to configure your server php settings. In the screenshot below, you will see the red On next to Display Errors, this is ok so you can move on to the next step as this was just my own system displaying this.

License Acceptance - I know most do not read these but you may want to at least once. To continue, read through the license terms and if you are ok with it, you can move on to the next step.

Database Configuration - The next step is where you tell the installer about your database. There are 7 settings to configure:

FTP Configuration - Setting up your FTP configuration was generally a needed function to allow a smoother installation of extensions once your site was up online. However, this is usually not required anymore which is why this step of the install process makes it optional for you. Try it without this information being filled in and see if it works for you; you can always fill this in from the admin global configuration section later.
NOTE: If you do fill in this information, make sure you also use the FTP host address that is associated with your host account...it's best not to use the default settings that you see in the screenshot for this.

Main Configuration - Almost done! Now you can title your website and setup your administration access credentials (username, password, email). For the "Install Sample Data", I recommend ignoring this because I personally feel this is more of a distraction when it comes time to begin creating your website content categories, menus, and articles; I like a nice clean install.
NOTE: For your username, make sure you use something unique and not something common like "admin' or anything that would diminish security. The same goes for your password, make it difficult!

Finish - You are all done with the installation and all you need to do now is click "Remove Installation Folder" from your site files. Leaving this in becomes a security risk, so it's best to remove it. Once you removed it, you can then click on the upper right button "Administrator" to log into your admin area, or you can choose to view the "Site" front-end.

Since writing this article, I see Joomla is still developing and moving ahead with the next step in the latest version of Joomla, but I know it won't stop at just 1.7, so I will keep you up to date. However, now that Joomla 1.7 is pretty much out, the installation process, as far as I can see, is still the same as Joomla 1.6 and information provided here will still apply.