How to interact with them
Press <X> when standing next to them or press <MENU> and select the character progression icon at the top-right of the screen with the <LEFT ANALOGUE>, before pressing <A>. Then select the relevant gang and double click their portrait. You may also pledge to a contact to increase your contact standing.
Things you can do with them
You can increase your rating through contacts. By performing action district missions, you will gain contact 'standing'. This 'standing' is akin to an 'experience bar' in most MMOs. This determines how much a contact favors you, and what they will offer you, such as leased weaponry, cars, modifications, and more.
I've completed a contact, now what do I do?
Great to hear! You can now begin the contact's daily missions which reward Joker tickets. This is accessed through the same area you pledge to contacts at, and you may even find a rare reward through working for some of them occasionally! Be sure to check contacts constantly for new and exciting missions.