An artificial neuron receives information from multiple sources, but it doesn't treat all inputs equally. Some signals are crucial, while others are just noise. The mechanism for telling the difference is the weight.

Think of a weight as a knob that controls the volume of an incoming signal. A high positive weight amplifies an input, signaling that it's very important for the neuron's decision. A weight near zero effectively mutes the signal, and a negative weight means the input actually works against the neuron firing. By adjusting these weights, the network learns to pay attention to the most relevant features in the data.

Before a neuron can make a decision, it calculates a weighted sum of all its inputs. This process is a straightforward piece of linear algebra called a linear combination. It combines every input with its corresponding weight, then adds a final tuning parameter called the bias.

Each input ($x$) is multiplied by its weight ($w$). The neuron then adds up all these products. The final result, $z$, is then passed to the activation function, which determines the neuron's final output signal.

For example, in an image classifier trying to identify handwritten digits, an input $x_1$ might represent the intensity of a pixel at the top of the image. If the network is looking for the number '7', a high value for $x_1$ is good evidence. The network might learn a large positive weight $w_1$ for that pixel. Conversely, a pixel in the bottom loop of a potential '8' would be evidence against it being a '7', so its weight might become negative.

The weights and biases are the only parts of the neuron that change during training. They are the learnable parameters. When a neural network "learns," it is iteratively adjusting its millions of weights to minimize prediction errors. This is usually done through a process called backpropagation.

A trained set of weights is the compressed knowledge of the network. It represents all the patterns, relationships, and nuances extracted from the training data. In this sense, the weights are the network's long-term memory. They hold the generalized rules for how to map inputs, like pixels in a photo of a dog, to outputs, like the label "dog".

The initial values of weights are often set randomly. During training, the network makes a prediction, compares it to the correct answer, and calculates an error. This error signal is then used to make tiny adjustments to each weight, nudging the network closer to the right answer. A weight that contributed heavily to a wrong answer will be changed more significantly than one that had little impact. Over thousands or millions of examples, this process fine-tunes the entire system.

Understanding weights is fundamental. They are not just numbers; they are the distilled wisdom of the network, encoding the importance of every feature to solve a specific problem.