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The word "weight" sounds technical. It isn't. You already use weights every day. You just don't call them that.
Imagine you're deciding whether to take an umbrella. You check three things: the weather app, the sky outside, and a text from your friend who is almost always wrong about the weather.
You don't treat those three the same. The sky outside gets a lot of weight. The weather app gets some. Your friend gets almost none, you've learned not to trust them.
That's a weight. It's how much you trust a signal. How much you let it push your decision.
In a neuron, weights are numbers that do the same job. A high weight means: "I take this input seriously." A low weight means: "I barely listen to this." A negative weight means: "This actually pushes me in the other direction."
That's all a weight is. A measure of how much something matters.