Of all the characteristics of a bike's tire, grip is the most important, as it defines the ability of the tire to brake hard without the wheel locking, of accelerating without loss of grip and of negotiating bends and changes in direction without skidding. In other words, it is synonymous with safety.
Rubber is, of course, the main component, but there are also many other compounds or chemical elements, such as silica, for example. The exact composition of tires is an industrial secret which is closely guarded by manufacturers.
One usually talks about "soft rubber" and "hard rubber" to distinguish the grip of different tires. A soft tire gives intrinsically more grip than a hard tire.
Each tire, and thus each chemical composition, has a corresponding range of optimal operating temperatures. Each tire has maximum grip at a given temperature or range of temperatures. Grip increases with the tire's temperature until maximum grip is reached. Grip then decreases (often at the same time as the tire itself deteriorates) if the temperature continues to rise.
The temperature of a tire increases depending on the mechanical stress to which it is subject: driving, accelerating, braking and taking bends are all actions which cause the tire's temperature to increase. As a rule, soft tires have an operating temperature which is higher than a hard rubber tire and take longer to increase in temperature.
Besides the specific grip achieved by each model of tire at a given temperature, one must also bear in mind that grip also depends on the surface of contact with the ground and the state of the surface being driven on. The greater the surface of contact between the tire and the surface, the greater the grip. If one takes equally wide tires, a tire without tread or with little tread thus gives better grip than a tire with a lot of tread. This, however, is only true on a dry surface.
In fact, on a wet surface, a large tire with little tread runs a greater risk of aquaplaning. The phenomena of aquaplaning is caused by loss of the tire's grip on the road. A surface of contact which is reduced to zero is equivalent to grip which is reduced to zero. The role of tread is to evacuate water and to stop a wave from forming at the front of the tire - the cause of the phenomena of aquaplaning.
Each type of tire has a speciality. One must therefore define the use and conditions of use of one's bike in order to choose the right tire.
For example: a "sport" tire will not necessarily be the tire which gives the best grip if the bike is mainly used for short urban trips in a cold and wet climate. Even though, in theory, the "sport" tire gives maximum grip which is far greater than a road tire, it will never have the opportunity to achieve maximum grip because it will never be able to reach its optimal operating temperature during short urban trips.
On a wet surface, a "sport" tire will run the risk of aquaplaning due to its relative lack of tread. For this type of use, a "road" tire would be more suitable. As its operating temperature is relatively low and quickly reached, it will operate in its optimal temperature range and will thus give greater grip than a "cold" sport tire. Moreover, on a wet surface, it will reduce the risks of aquaplaning due to the size of its tread.