First, make sure that you're looking at a record for a print book (not an ebook) or article. Libraries usually aren't allowed to lend electronic materials to each other.
Click on the title of the item you want:
Look under "Availability/Holdings" for the "Request Item" button and click it (don't worry about the list of libraries that have it; that's our job).
On the next page, scroll down a bit to the contact information form and fill it in; then hit "Submit."
You will then be taken to a confirmation screen that informs you that "your request has been placed." You will receive an email from the library when the book arrives and is ready for you to pick up. This typically takes 1-2 weeks for books. Articles often come within a few days and are usually sent to you as email attachments.
If you are asking for something that may be rare (such as an older item, a limited edition, or something in a foreign language), you may want to look at the list of holding libraries just to make sure that several libraries own the item you need. If a book is only held by two or three libraries in the world, the odds of our being able to borrow a copy will be low and you may need to find a source that is more widely-held.
For a video demonstration, visit https://youtu.be/zvs_144F5Ok.