Final answer:
Dendrites are the branched structures of a neuron responsible for receiving incoming signals from other neurons, connecting at synapses through dendritic spines.
Step-by-step explanation:
The multiple thin, branched structures on a neuron that are specifically involved in receiving incoming signals are known as dendrites. These structures are crucial extensions of the cell body, or soma, and they play a fundamental role in the neuron's ability to receive and integrate synaptic input from other neurons. Each dendrite can branch out substantially, forming a complex array that resembles a tree, greatly increasing the neuron's capacity to make synaptic contacts. The dendrites' ends, called dendritic spines, also enhance the possibility of forming connections with other neurons' axons at specialized sites known as synapses. Furthermore, while neurons often have multiple dendrites, it is worth noting that not all neurons contain dendrites, and their numbers can vary based on neuron type and function.