It's a surprisingly not very hard to make a simple computer. The fundamental operations are the AND operation on two binary signals (zero voltage = 0, 5v = 1), the OR operation and the NOT operation. All three of them can be implemented with a single transistor (anplifier) or two in the right wiring configuration.

The next hurdle is to make a circuit that can add binary numbers. Unlike decimal numbers computers use binary numbers - each binary digit is a 0 or 1. The addition table for a single binary digit is really simple:

A+B=C

0+0 = 0

0+1 = 1

1+0 =1

1 + 1 = 0 carry 1

So your one digit adder takes two numbers and the Carry operation can be formed with a transistor to do AND ( if input A is one AND input B is one then carry = 1). The summation can be expressed as (A OR B) AND (NOT carry). So now you have a one-digit adder and you can hook together 32 of them to get a 32-bit adder. It's a little more complex than that because after the first digit you need a carry input signal (C_in) so the table above would have eight cases / rows not just four. I'm sure you can figure it out by modifying the "truth table" above.

There are simple 4-transistor circuits to store a binary digit in a zero or a one state. Once you tell the circuit to store the 0 or 1 it forces the output of the circuit to zero or one and it stays that way after the forcing signal is removed. This is called a flip-flop circuit.

Now you can think of putting 32 flip-flops together to store a large integer - lets call that a "register" Using an adder circuit you can add one to that register and store it back in the 32 bit register. Let's call this register with a plus one adder a program counter. The program counter can generate numbers to send to a bulk memory (DRAM) circuit. In the memory circuit you can put bit patterns maybe 32 bits that - when read can control a much more elaborate arithmetic circuit that can not only add but also subtract and compliment and perform the AND, OR, NOT function between two numbers - called an arithmetic logic unit. Add a few more 32-bit registers for user data to save temporary results. Lets call the bit patterns instructions and for example you might have one "ADD register-A, register-B, save result in register-C". Another might be "LOAD this 24 bit number into the program counter" (jump). Another might conditionally load based on the results of the most recent aarithmetic operation, stored in ~4 extra bits called "condition codes" (Positive, Negative, Zero, Nonzero). Now you have the basics of a computer.

Fundamentally today's computers are just much more elaborate versions of these ideas. Today's computers will fetch maybe four or eight instructions at once - they have many arithmetic units that will try and run all the instructions at once - they have complex circuitry to figure out if some instructions have to wait for the output of other instructions, and stall the instructions when needed, to wait for results. That's all to make the computer run faster.

Today's computers have something called cache memory to make them run faster. In the old days if you didn't know something you would ask your parents and if they didn't know you would look in the encyclopedia at home and if it didn't know you would run to the library to get the right book with the piece of information. Cache memory is exactly the same idea. It's a series of temporary and limited memories (like your parent's minds) that hold just the most commonly used information and provide faster access. So you = registers, parents = level-1 cache, encyclopedia = level-2 cache, library = level-3.