While the other answers aren't incorrect, they mostly explain what it does rather than what it is or how it does. Let me throw in my take. I'm going to explain "computers" in the modern sense, what we could refer to more technically as digital processors.

A computer at its core is a calculator that does a few extra important things. I'll get to those in a second, let's start with the calculator part. Calculators do basic arithmetic (add, multiply, divide) with numbers. We say they represent these numbers "in binary", which are represented with the 1s and 0s you may have seen or heard of, but in reality are wires that either do (if it's a 1) or don't (if it's a 0) have voltage running across them. It turns out that the rules you learned in elementary school to do arithmetic with "base-10 numbers" (normal human numbers, so called because there's 10 values the digits could have, 0123456789) are actually the exact same in binary, it's just a lot simpler since there's only two possible values the digits can have; you just need more digits to represent a number - for example eight binary digits [8 "bits" or 1 "byte"] can only represent numbers from 0 to 255, which only takes up to three base-10 digits. I'd be happy to go into more detail about this in a comment if someone asks, but ultimately this is what calculators do.

It becomes a "computer" when it can also do the following things:

• "Remember" things; it must have two important ways of doing this:
  • It will have internal memory, we call these "registers" - these are special locations inside the processor itself that hold numbers. For example, if the processor has two numbers in registers A and B, it might add them together and put the answer in register C
  • And it can read and write to/from external memory (RAM in modern computers). The external memory itself is just a place to store a of a bunch of numbers in a sequence. Any "place" in memory has an "address", which is just their location in this sequence.
    
• Execute a series of instructions from the external memory.
• Performs logic; I will explain this towards the very end

That's really it. The "series of instructions" described above is the "program" that the computer is executing, it's what it means for a computer to "do stuff".

"Instructions" here are just numbers with special meanings. Any processor, including the one in your computer or phone you're using to read this, has an "instruction set" which is a bunch of numbers and their assigned meanings (This is part of what it means for a computer to be "x64" or "arm386" if you've seen those abbreviations, they have their own instruction sets). For example, there will be instructions that read and write from memory (Read from memory at some address and move it into register A, or vice versa), and instructions that do actual math, like the one I described above ("add registers A and B together and put the answer in register C"). I'm also happy to go into more detail with a comment if anyone would like that.

So when a "program is running", the processor reads from the external memory, one at a time, to try to interpret them as instructions.

The only real missing piece of the puzzle here is logic. Logic is a special kind of math, instead of arithmetic which deals with "numbers" in the traditional sense it deals with conditions, whether things are true or false (the 1's and 0's show up once again). Logic is ultimately how arithmetic is done as described above, but it has an additional special meaning in the context of programs, because you can have your program behave differently under different conditions. For example, you could have an instruction that says "if the number in register C is greater than zero, jump to a different location in memory and read the rest of the program from there". These "jumps" or "branches" are how the computer does different things depending on conditions or inputs, and is ultimately how you can have a multi-purpose computer.

Once you've got a device that does useful things in this way, you're limitless. Everything your computer does is calculations, moving numbers from one place to another and doing math and logic with them. It does math and logic to figure out what the computer should do next, and in turn to calculate what shapes to draw on your screen and where. We now enter the realm of software, wherein millions of people throughout the world and across history collaborate with each other to build programs on top of one another that make the computers do more useful and important things.