Measurement is the act of extracting information about something. Weighing something on a scale is a measurement. Touching something and feeling a force back is a measurement.
This is a deeper concept than it seems. Fundamentally, all information we know about the world was measured in some way. We only know what we can measure. Measuring is knowing.
We are measuring all the time
All the concepts we have: “object,” “particle,” “thing” are just words for patterns we have found in measurements. Indeed, seeing is a measurement: we are measuring the photons that enter our eyeballs!
Usually, with physical objects like balls, we are always measuring. We are touching it, or photons are touching it and hitting our eyes. When we aren't measuring it, something else usually is. Air molecules are hitting it and reacting.
When nothing is measuring
However, there are some things, like electrons, that are small enough to avoid constant measurement. So, scientists in the early 1900s started asking an odd question: do small objects like electrons have definite properties (like position) when they are NOT being measured?
The answer was, shockingly, no. Therefore, physicists invented quantum mechanics as a way to describe the properties of an object when nothing is measuring it. Between measurements, electrons do not have a single, definite position. They exist as a blend of possibilities. The act of measuring is what forces the object to commit to one definite answer.
A simple example: electron spin
To illustrate this concept, I'll show it to you in a very simple, controllable quantum system: the electron spin. Of course, there are many quantum properties in the world: electron position, electron velocity, light wavelength. But all of those are very complex, and it's easy to get lost, because each of these has infinite different possible measurement results! For example, we can measure an electron's position and find out that it is on your finger, or your laptop, or anywhere in between.
In contrast, the electron spin is what we call a binary state: when we measure it, it can only be up or down, but never in-between. This was found in the legendary Stern-Gerlach experiment of 1922. Therefore, we'll work with that one. It makes the concepts a lot simpler.
Below is a single electron in its spin state. You can do two things to it: fire a laser at it to drive it between “up” and “down,” or measure it and read the result.
Hold the laser to drive the electron continuously between “up” and “down.” The probability of each outcome is set precisely by how long you held the laser, even though the measurement result is still random. The in-between state is well-known. It is just not measurable until you measure it.
Superposition
Superposition is the name for that blend of possibilities that exists between measurements.
Before we measure it, an electron can be “mostly up,” or “half up and half down,” in a way that has nothing to do with our ignorance. It is not that we don't know the spin yet. It is that there is no single spin yet. Measurement is what creates one.
A useful image is a spinning coin. Mid-flip, the coin isn't heads or tails. It is a real, physical combination of both, and the moment your hand slaps it down is the moment it becomes one or the other. Quantum superposition is similar, except the “flip” is the natural state of an electron whenever nothing is touching it.
This is what gives quantum computers their edge. A regular bit picks 0 or 1. A qubit can hold a superposition of 0 and 1, with specific weights for each. Run many qubits together and they explore many combinations at once.