A talk page is a discussion page that you can find attached to some article and user pages. They exist for communicating with other users and discussing how to improve the wiki.
To get to a talk page from an article, just click the "discussion" or "talk page" link or button near the title of the page. To get to a talk page from a profile, just click the "Talk page" tab.
To start a new message, click the "Leave message" button at the top of the talk page.
Enter your message in the editor window. At the end of your message, type four tildes ("~~~~"). This will convert to a signature when you hit save.
Enter the title of your message in the "Subject/header" text field near the Publish button.
Click the Preview button to check your work.
To respond to a talk page message, simply edit that area of the talk page, and indent your reply. You can indent by putting a colon (":") at the front of each line. Be sure to sign your response with ~~~~.
Keep your comments polite and friendly. It's always best to be civil, even if things get heated.
Archiving talk pages
Archives of talk pages are made when a talk page becomes too long for either the user to easily find a past conversation or for one's web browser to render the talk page in a timely manner. An archive is simply a subpage of the user talk page where old conversations are stored (e.g. [[User talk:Foo/Archive]]).
To create an archive, simply cut (Ctrl + X) all the old conversations from your talk page, and then saving it to show the reduced size. Then, create a new page in your user talk space and paste (Ctrl + V) the old conversations there. If you already have one archive, create a second archive page (e.g. [[User talk:Foo/Archive 2]]) to avoid having the the same issues that cause you to create an archive in the first place affect the archive page itself. You may also consider using the Archive Tool for auto-archiving the pages.
When archiving old discussions, it is customary to leave current, ongoing discussions on the existing Talk page. In some cases, a header text is also preserved. Also, it is good to leave a link near the top of your talk page to the archive so users can easily find previous discussions.
It is recommended that you do not attempt to archive another user's talk page. These are not subject to the recommendations on this page, and each user may choose alternate means of archive their own talk page, including choosing not to archive at all, but to instead remove old messages. Though since those messages are other users' contributions, this is generally rude to do and in bad taste, with archival being more polite and considerate of what others say.
When archiving your talk page by moving it, be aware that this causes the newly-created archive to be added to the watchlist of any user who was previously watching your talk page. This could be annoying for people who have no desire to watch your talk archives.
What if Talk Pages are unavailable?
For Articles: If a talk page is unavailable on an article page, you can usually find comments at the bottom of the page for use instead.
For User Profiles: If a talk page is unavailable on a user profile, you can usually find a Message Wall tab in the same location for use instead.