The first most significant feature is a tread. You can have the stickiest compound in the world, but if you have no tread, wet weather performance will be rubbish. As you know, tread patterns break up the layer between the rubber and the tarmac so that aquaplaning is discouraged.
As a tyre rolls over wet tarmac, a 'wedge' of water forms in front of the tyre. If the tyre rises up on this wedge, contact between the tarmac and rubber is broken, and aquaplaning occurs. By breaking up the surface of the tyre such that there are channels for this water wedge to infiltrate, the tyre is less prone to rising up on the wedge. For a similar reason, (like-for-like) narrow tyres perform better in very wet conditions, - the tyre is less prone to 'floating' up onto the wedge of water.
More detail: Circumferential grooves (going 'around' the tyre) act as channels to drain the wedge, and to some extent, store this water. Lateral grooves, - going side-to-side across the width of the tyre - drain this water from the circumferential 'grooves' to the tyre edges. At 100km/h, an average tyre in average rain needs to shift something like 6 litres of water per second to maintain contact. New wet tyres have massive grooves in the centres which can displace huge amounts of water, - something like 10 litres per second, but these wear very quickly in the dry. Circumferential 'ribs' provide contact zones, particularly for cornering. Small, shallow dimples around the shoulders of the tyre increase surface area to allow for more efficient cooling. Sometimes a 'void ratio' is specified, which quantifies the amount of non contact/ contact area across the width of the tyre, - for a slick tyre this ratio will be 0%, for an off-road mud or snow tyre this will be much higher, perhaps 40%?
More detail still: Tread patterns vary for a number of reasons. Simple block-shaped treads fundamentally work well in muddy and wet conditions, but are prone to rapid wear of the front and back of the block on dry tarmac. Regularly spaced grooves are avoided because they set up harmonics which can cause hefty vibrations.
Finally, asymmetrical tyres will have more large blocks on the outside for cornering stability, and smaller blocks on the inside for heat dissipation and water displacement.