How is information sent from the skin to the brain?
The accuracy of sensation is actually quite variable depending on the region of our body. Highly innervated areas such as our fingers have a higher degree of accuracy than sparsely innervated areas such as our legs.

There is a simple experiment you can do to illustrate this. Close your eyes and then get a friend to lightly and slowly run their finger from your wrist, up towards the bend in your elbow on the inside of your arm. Try to guess at what point the reach the joint of your elbow. 

In terms of how the brain knows where it came from,  there is part of your brain called the sensory cortex. This sensory cortex has a pattern that corresponds to areas of the body. This pattern is known as a the sensory homunculus (google images will find this). As signals pass up the various different spinal pathways and into the brain they are sent up to this sensory area and the appropriate region of the homunculus is stimulated thereby giving the sensation in the appropriate body area. 

Interestingly in some brain surgery a patient may be kept conscious so that these areas can be directly stimulated by electric current and the patient will experience sensations. 

There is also a matching area, called the motor cortex, which corresponds to movement rather than sensation. 

The above is a simplified version of what goes on and there is lots more detail which would be covered in neuroanatomy books