As long as you have not changed any core Joomla! files, you can update Joomla! pretty easily.  All of the content you create in your Joomla! site is stored in the database, so updating to the latest version of Joomla! will not erase your content.

Updating from Joomla! 1.0.x to 1.5.x is considered a migration and is much more involved.  It requires special tools that can be found here:

http://docs.joomla.org/Migrating_from_1.0.x_to_1.5_Stable

This article is only intended to help upgrade between minor versions of Joomla! (example: 1.5.9 to 1.5.15).

The Tools You'll Want

There are tools out there to help make the process easier and safer.  We use the steps and tools below to keep our sites up to date.

Step 1: Backup Your Site

The first step is to make a backup of your site.  For this, we suggest using a free component called JoomlaPack.  Backing up is just good practice, and we recommend doing this regularly whether you are upgrading or not.

http://joomlapack.net/

Step 2: Check Files For Custom Modifications and Differences

In case you have modified core Joomla! files (or are unsure if someone else has), you can use the free Version Verification Tool component to see what Joomla core files are different on your site:

Get the Version Verification Tool from CMS Market here

The tool compares MD5 checksums of the files on your system to the original MD5 checksums to determine if files have been modified.  If files have change, it will try to determine which version the file is from.  If not version can be determined, it can be assumed that it is a custom modification.

Note: If you have many files on your system with unknown versions, see the article "See How Files Have Been Modified On Your Site" to help determine how the files have been modified.

Step 3a: Update To The Latest Version

There is a free component called Update Manager that will automatically get the latest Joomla! package and unzip it on your system:

Read More about the Update Manager

I have also been told about a commercial component called Magic Updater that does the same thing but comes with a few other bells and whistles like backing up files and updating multiple sites from one install:

Read More about Magic Updater

Step 3b: Update To The Latest Version (Alternate/Manual Installation)

Or you can download the packages from Joomla! and install it manually:
http://www.joomla.org/download.html
The download page has the location of the package file you will need as well as a link to these instructions on how to manually upgrade your Joomla! site:

http://docs.joomla.org/Upgrading_1.5_from_an_existing_1.5x_version

That's All Folks

Though it seems daunting at first since there is always concern you can break something, if you follow these steps, have a backup (just in case), and you'll be fine.

If you have other tools or tricks that you use to make upgrading easier, let us know and we will incorporate it into this list.