Connective tissue. I believe you post is looking for either 'connective tissue', or fascia. Each joint has plent of tendinous/ligamentous support to prevent dislocation; it is when these tissues develop laxity that the dislocation occurs.
This depends on what. Joint you are talking about but in general, muscle strengthening exercises cn help to stsbilize a joint and if the joint is a weight bearing one, lose weight if overweight to reduce the stresses born by that joint. Some joints due well with bracing or tapng. Other than surgery, there is nothing that directly reinforces the joint capsule although there is ongoing research with various leads.