Summary:

Intuitively a climber's hands play a crucial role in maintaining stability and upward movement on a vertical surface. Many times, the friction between a hand and a bad hold is very capable of being lost which can result in the overall loss of balance on the wall and will culminate in a fall.  Unlike the progression of climbing shoes that have increased in performance advantages, the only tool climbers have for their hands is a largely ubiquitous- chalk. 

Skin

Skin is an important variable climbers attempt to control. The interaction our hands have on a rock surface can be summarized by
​P = F/A
P = Pressure
F = Force
A = Area
The skins surface have contours called friction ridges(increasing surface area) that lend to better grasp and thus the ability to hold onto smaller or more awkward holds. 

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Temperature

Hot
In hotter environments, the sweat produced by a climber's hands acts as a natural lubricant which is detriment to being able to firmly grasp climbing holds. For this reason, climber make use of chalk to absorb the moisture and dry their hands.
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Chalk Composition: climbing chalk is largely just a magnesium carbonate compound with sometimes an added component for drying the hands. While studies have actually found that chalk lowers the static coefficient of friction compared to without it is still argued to produce better friction than with sweaty hands. 
Cold 
Colder more dry climates are much preferred to climbers. Two effects occur when a human's hands get cold: they produce less sweat, and the skin becomes harder as the cold reduces fluidity of the biological components of cellular membranes. For these reasons hand friction is much better when the body is at colder temperatures and results in stronger adhesion to the rocks surface.

Rock Composition

The type of rock is also an important notion to consider as the same hold on two different types of rock may feel incredibly different. This for several reasons:
Grain Size and shape: the Crystal structure of different rock types vary with how they formed over millions of years which leads to different amounts of apparent friction coefficients. 
Porosity: referring to the proportion of non-solid volume compared to total volume of material (i.e. the rock is lumpy and jutted with unfilled volume around those lumps).
Well Cemented: In some rock types the grains will gradually break off over time and can act like ball bearings for your hands and feet drastically decreasing friction.