You should make sure that you only copy the files from inside this directory, rather than replacing your entire wp-content directory.
This is where your themes and plugins live, so you will want to keep them.
To perform the update, you just need to fill in the connection credentials for that FTP account.
Whether your files are owned by the web server user, or not, will depend on how you installed Word Press and how your server is configured.
This includes all the core files used to run Word Press.
Once you are on the "Update Word Press" page, click the button "Update Now" to start the process off.
If you see a "failed update" nag message, delete the file .maintenance from your Word Press directory using FTP. If the one-click upgrade doesn't work for you, don't panic! These are the short instructions, if you want more check out the extended upgrade instructions.
If you experience problems with the Three Step Update, you may want to review the more detailed upgrade instructions For these instructions, it is assumed that your blog's URL is directories and sub-directories, and in the root directory (such as index.php, and so on). Be careful when you come to copying the wp-content directory.
For Word Press 3.7 , you don’t have to lift a finger to apply minor and security updates.
Most sites are now able to automatically apply these updates in the background.Otherwise, visitors to your site (including you) will continue to see the old version (until the cache updates).