The bigger the light, the softer it is. The smaller the light, the harder it is.
“Soft” or “Hard” light refers to the transition area between what is being lit, and what is shadow. Hard light has sharp lines for it’s shadows, like the midday sun, or a puppet show. Soft light is ‘blurry’, and transitions into shadow slower.
Soft light also brings out less texture detail, and tends to light things more evenly. It is considered great for portraiture, as it can be very flattering.
Hard light is often considered dramatic, due to it’s ability to bring out texture and detail, and cast shadows like in noir films.
Perceived size, not just size.
You could have a light source as big as the sun, but if it’s really far away, then it’s still small to the subject, so it’s going to be a hard light source. Like… the sun. The sun is a pretty good example of this.
This is why photographers tell their accountant that they need such massive studios. It’s not just because they like high ceilings, but it’s so they can back their lights far enough away in order to get the look they want to get.
In image A, below, we are lighting the subject with a single bare flash. (and the window, technically. Ignore that, that’s just so we can see the room for context) It’s a very small light source, and would produce hard shadows. If we want to soften it up, we can put a shoot-through umbrella and fire the flash into that. The umbrella diffuses the light, and now we can see the light source in B is much larger.**
If this is too large for us, the shadows a bit too soft, what can we do? We could move it further away! Like in C. What if we want much softer light? We can move the umbrella closer, as in D, and get nice soft shadows.