Logic gates allow the processing of information in the form of voltage differences using boolean logic. For each gate the voltage potential at the input(s) dictate the output voltage as according to the truth table for that gate. My favorite gates are the Universal NOR and NAND gates.
They are universal because these gates can be chained together in various ways to build any other gate. These two logic gates are thus completely self sufficient in their ability execute any mathematical logic that exists.
One of the most natural extensions of logic gates is the ability to store information as the presence or absence of a voltage potential. In this case a 0 volt potential represents a boolean 0 and a high voltage potential (i.e. > 4.5 volts) represents a boolean 1. A single digital output is called a bit. One of the simplest forms of storing a bit is in the flipflop circuit.
When attacking a big problem I always appreciate following the reductionist mindset. Generally, the simple thought of building a computer from just wires and IC’s seems completely mental. However, if we break the problem of assembling a working computer into parts, i.e. sets of simpler problems. Then all we have to do is solve all those, and bring them together. From that perspective building a computer is no harder than building anything else. It simply requires resources, experience and a lot of patience.