Tire tread design varies quite a bit in appearance, but most passenger vehicle tires are designed to channel water away from the contact patch to achieve the most grip.
Slick tires have no tread, so they are very good at gripping dry paved surfaces, but have no way of channeling water away or clinging to loose surfaces, so they are a very purpose-built tire design. Slick tires are not street legal in many countries, due to how unsafe they are in wet road situations.
Off-Road Tire Tread
Tire treads designed to go off road have treads which grip loose surfaces well because the material can get between the threads, and the force of the wheel pushing the material together gives the tire something to cling to.
Snow Tire Tread
Tires designed to perform well in snow are similar in concept to off road tires, but have additional gripping points due to ice being a very low-friction surface. The rubber compound is carefully considered because in the snow, the tire won't heat up (which softens the rubber in other tires, providing more grip) anywhere near as much as tires in other conditions.