Following someone adds them as a contact, which means you can post messages to their wall and track activity on their wall. The wall activity shows up on the Overview page, in the Recent Activity panel on the right, and in your daily email digest.
Note: You can't follow outside users (people outside your company), but you can add them to your contacts. Outside users' names appear in purple text, and they have icons () after their names.
For more information, see How do I follow people or add contacts?