Traditionally, robotic grippers are based on stiff materials, enabling end effectors with high load capacity and precision for industrial applications. Recent advances in soft robotics have led to a proliferation of novel gripper designs with increased conformability to accommodate objects of varying shape, size, material, and surface properties, allowing for grippers that can safely manipulate a wide range of objects. While compliant materials offer noted advantages for robotic grasping, their ability to deform limits their load capacity. Therefore, stiffness selection is critical in gripper design, and the use of materials with tunable stiffness can be exploited for new functionality. Here, we present a mechanics-based investigation of the design of versatile grippers that can accommodate both soft and stiff grasping modalities. We examine the ability to form contact and how different types of gripping forces, including frictional, normal, and adhesive interactions, can be leveraged and controlled. We use analytical models based on elastic beam theory and contact mechanics to quantify the relationship between gripper deflection, contact area, contact pressure, and load capacity. We then use these models to define quantitative conditions for successful grasping as a function of the geometry of the object and the stiffness and geometry of the gripper. Finally, we conclude with an experimental case study and a discussion of how stiffness can be selected and modulated to realize successful grasping for different classes of objects.