Basically it has formed in such a way as to correctly associate the impulses it receives with the triggers. And yes there are individual stands of nerves that take certain paths. It doesn't always work. Pain that should be associated with some organs ends up feeling like it comes from somewhere else. There is phantom pain from lost limbs. You can even reprogram it to some degree. People can control and sometimes "feel" prosthetics via their chest muscles for instance.

It's not a different signal, but it comes in on a different nerve.

You don't have just one nerve that senses your whole arm. The nerve that senses a touch on your shoulder is separate from the ones that senses touch on your elbow, or your fingertips, or anywhere else on your arm. Your brain knows where you were touched because of which nerve gets activated.

Most of the responses here have already hit on the important parts so I'll just add a little extra.

When you sense touch, that signal is sent back to the spinal cord and then up to the brain. That signal is interpreted by your sensory cortex which has a map of your entire body called the homunculus. So when your shoulder is touched, the some of the neurons in the shoulder part of the sensory homunculus get activated which your brain can then interpret as a touch on the shoulder. Scientists have seen that the distribution of neurons is not equal to the surface area of your body parts. Just because you have more shoulders than hands doesn't mean you have more neurons dedicated to sensation for your shoulders than hands. In fact, its quite the opposite. Your hands, particularly the fingers, have a high density of sensory neuron endings making them exquisitely sensitive to touch. Your shoulder (and most other skin) has much fewer nerve endings and is less sensitive and thus takes up less of the sensory homunculus. This is why you can feel fine details with your finger tips but not with your shoulders or even the back of your hand.

Pain is interesting since there are pain sensing neurons in for both your somatic nervous system (skin, muscle, things under conscious control) and autonomic nervous system (stomach, heart, things under unconscious control). We are not very good at localizing pain coming from our autonomic nervous system which is why, for example, stomach aches often feel like a more generalized pain/unease. This phenomenon is also why appendicitis will first present like a stomach ache or cramp (autonomic pain) before eventually becoming sharp/burning pain in the lower right quadrant as the inflamed organ begins to irritate the skin above it (somatic pain).

Sensory homunculus - https://www.ncbi.nlm.nih.gov/books/NBK549841/

Somatic vs autonomic pain - https://journals.physiology.org/doi/full/10.1152/ajpgi.2000.279.1.G1

The signal is the same; it’s the starting and ending points that differ.

The touch activates local sensory neurons, which generate an action potential. This signal is transferred to the somatosensory cortex of the brain (soma = body, literally the body sensing part of the brain). There is point-for-point correspondence of each area of the body to specific points on the somatosensory cortex. This is called somatotopy, and is how the brain knows which part of the body is being touched.

Read about it here.

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Each patch of skin is supplied by a single spinal nerve and this is called a dermatome. When you touch that area of the skin that single nerve receives an impulse and takes that impulse to a specific region within your brain where it interprets it as a sensory input (in this case touch) from that region. Obviously it works on an even smaller level than dermatomes as we can distinguish touch from two nearby areas and this is because impulses are travelling up specific neurones which form the spinal nerve