Installing WordPress has their famous 5-minute install process; a process that only takes that long or less for the average user. If you have already seen how to install Joomla, it's similar in a sense of creating your database, uploading your files to the directory you want your blog site to exist, and then using your browser to navigate to this location to begin the installation.The overall process is easy and quick.
Before anything, you need the files, so you will need to visit the WordPress.org website to download the zip file (at the time of writing this, WordPress is version 3.1.2) to your desktop. Once done, you can unzip the file (again to the desktop location) because you will need to make a modification to one particular file named "wp-config-sample.php" which you can see highlighted below:

You will need to create a single MySQL database, because without one, your website won't function. Your installation will depend on your version you are installing because right now, coming in June, 2011, WordPress will be version 3.2, so I've provided you with server specificaton requirements from wordpress.org for both the current and upcoming version:
WordPress server requirements for Version 3.2 (June 30, 2011):
WordPress server requirements for Version 2.9, 3.0, 3.1:
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.
In your WordPress folder, you should see a file named "wp-config-sample.php" which you will need to open it up in your favorite html editor. Generally html code editors will display your code in colours so you know what parts are what, but some can simply open this file up in NotePad (a Windows text editor) to make the necessary changes. I will show you the basic setup of this file using my Dreamweaver:
Configure - Adding your Database Information
My Dreamweaver displays php code in the colours that you see here and easily defines the areas to be edited, the lowercase red text between the single quotes:

The next thing you need to do is to rename this file from:
wp-config-sample.php
to this name:
wp-config.php
You are now ready to upload your complete WordPress structure of directories (folders) and files. I prefer to use an FTP program to upload files because it gives me more control and also ensures it's done correctly. Some hosts provide customers with a feature known as "Fantastico" which is basically an almost instant auto install of a favorite script, whether its blogging scripts like WordPress, forums like phpBB3, or other forms of scripts. From my experience, I see a lot of people who run into complications, so I prefer using an FTP program for file management, like WS_FTP...although there are quality free versions like Filezilla.
When your files are uploaded, you can now use your favorite browser to go to the location your WordPress is located and follow the instructions, starting with the "Welcome" window:

Seeing this screen in your browser means you were successful with your file upload and wp-config.php information.
If you see an error like this:

...you will need to double check your database information in your wp-config.php file to make sure everything is correct. Although this is the most common reason to get this error, it might be something different, such as the database not being setup properly or even a permissions issue. You may want to contact your host support if you continue to run into problems if you are sure your settings are correct.
NOTE: For a more secure administration, I recommend NOT to use the default "admin" username for your site (or any site) because it makes things easy for those who wish to try to gain access. I suggest creating something unique and difficult and not use actual words or names because even this makes it easy.
Installation - Success!
If all went well, you should see your successful install screen with your secure username and password. Please note what you see in the screenshot is just for demo purposes and your own username and password will show up instead:

Quite true, I have done a number of WordPress installs on sub-domains where I set up test sites and when I have finished and uninstalling them I have had trouble with Fantastico as it doesn't always remove the databases and other files making a call to support necessary.