When you search for people, or view your connections page, you can view other people's profile. You can also click through and view their connection's profiles, if they have allowed this in their own privacy settings.
When you are viewing someone else's profile, you might have the option to Add them as a connection. If so, click this link to open the confirmation window, then click Add connection. Depending on their settings, they will receive an email notification, and be prompted to confirm your connection request.
For more information on connections, see Managing your connections.
Note: Depending on each person's individual privacy settings, what you see may be restricted to certain pieces of information. You too can control what other people see of your own profile in your privacy settings.