Case Contact role is a related list on the case detail page.
On case detail page we already have contact information. But if you want to associate multiple contacts with the case then we have to add Contact Role related list on case page layout. Click New and select the contact from Contact Object and set the role for this contact and save. The contact is added to the related list.
Setup > Customize > Cases > Contact Roles
Case Contact Role Picklist Values: You can create new, edit, reorder, replace, printable view case contact role.
Case Contact Role Picklist Values > New
Add one or more picklist values and each value create own line and save.
Other Contact has be added into Case Contact Role Picklist Value.
Contact Role Related List on Case Page Layout.