The most basic idea behind most modern computer chips is the semiconductor. It can be thought of as a sort of electrical valve, a substance which is either an electrical insulator or conductor until a small electrical current is applied to it at which point it changes into the other.
By having a substance that stops current flow until a signal allows it, or allows it until a signal stops it, you can perform logical operations. For example if you have a semiconductor which is normally an insulator you will have two inputs and one output. If only the input which tries to pass the semiconductor is given a current it is stopped by the insulation and there is no output signal. If only the input to the semiconductor is given a current the semiconductor becomes conductive but there is nothing to pass through so there is no output. Only if both inputs are given signals does a current pass through to the output, meaning it is effectively the AND logical operation.
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