I know how scary this can be ... take a deep breath, relax. Lots of people here that will help you through this.
So make sure you back up everything. Hopefully you have FTP access. If you do, copy all the files in your WP root directory and all the folders to your local hard drive.
Down load the latest copy of WordPress (link is at the top of all WordPress.org pages ^
Read the manual update process at this page.
There are some files to be very careful of. Make sure you don't delete them or your have copies of them,
wp.config.php and .ataccess files in the root directory
You can delete wp-admin and wp-includes directories and then re-upload them from the download package from here. I've found that deleting these 2 folders and re-uploading them often solves my problem with upgrade corruptions issues.
The one directory that is very sensitive is your wp-content directory. This dir has all your plugins and theme in it, so MAKE SURE you have backups for this directory before you go near it.
Within the wp-content dir is a folder called "upgrade" Rename it to "upgradeorig". Sometimes by renaming this dir, your auto upgrade feature will work again.
Study the docs (link above) and make sure you're thoroughly familiar and comfortable with the processes before you start.
Make a written check list and follow it as you progress through the steps.
Need clarification, ask away!