Why do clouds have well-defined boundaries? Why do cumulus clouds have well defined boundaries? In other words, what are the physical mechanisms that hold a cloud together, as an entity separate from other clouds, that prevent it from spreading, etc.
Naively, one could expect the atmospheric vapour to spread homogeneously or (in presence of nonlinearities inherent in hydrodynamics) form periodic structures or even vortices - these are indeed observed (see, e.g., horizontal convective rolls), but does not seem to explain the cumulus clouds. Dust clouds in outer space, held together by gravity seem closer phenomenologically, but physically gravity seems as a less plausible explanation than fluid/gas dynamics
 A: Cumulus clouds are opaque (optically thick). The interior of a large cloud is very dark (as you've experienced if you've flown through one in an airliner in the daytime during ascent or descent). When you view a cloud from outside, the white scattered light you see comes from the outer layer of the cloud where sunlight can reach.
Starting from the outside, as the droplet density increases heading into the cloud, the scattered intensity becomes greater and greater, until so much light has been scattered out that the remaining inward-bound light is significantly attenuated. Around this point, the scattered intensity reaches a maximum and then quickly dies out. Even if the droplet density continues to increase toward the center of the cloud, this does not contribute to the cloud's brightness.
(This is also why sunlit cumulus clouds are very bright -- they scatter nearly all of the incident sunlight, similar to an ordinary bright-white diffusely reflecting object, like a sheet of paper.)
When viewing a cloud from outside, you see mainly the "skin" at the depth of maximum scattering. Thus, you see a nearly uniform brightness on any line of sight that intersects this skin, despite potentially large variations in depth and density under the skin. As your line of sight starts to miss the skin, the brightness you see decreases along with the droplet density; if this occurs over a distance that is small compared to the size of the cloud, it will appear as a sharp boundary.
A: I just want to expand on Anders’s otherwise great answer, because I think based on the text of the question this paragraph is more what you're asking for:

There are doubtless other forces keeping cumulus clouds sharp, like the upper boundary often corresponding to the top of an upwelling convective flow into drier air and hence having a strong vapour gradient, and the cloud base being set by temperature, pressure and the dew point.

What the layperson needs to understand about a cloud, is that it is not like a rock with a defined set of grains/crystals in a fixed configuration. It is more like a river or a waterfall. Stuff is coming in one end and going out the other.
At the bottom you have moist air coming upwards, where the water inside is fully vaporized and so is transparent. But the pressure is dropping and so is the temperature, and at a particular virtual boundary which is the flat bottom of the cloud, this vaporized water condenses into little droplets. Those droplets are visible now.
All of the little droplets that you are seeing in the cloud are drifting, and on average they are drifting upwards.
Partly this is driven by outside factors, different masses of air moving across the landscape collide and one of them shifts upwards while the other one burrows underneath. But it is also self-sustaining. As a consequence of condensing, droplets release heat which warms the air which expands it. The warm air rises and drags the droplet with it upwards. Because the flows involved are way slower than the speed of sound, the air behaves approximately as an incompressible fluid. That just means that if there is this sudden updraft from the condensation warming the air, then air gets sucked in from the bottom to prevent a vacuum. (Incompressible also means not expandable, the vacuum can be interpreted as “stretching out the air” and at lower velocities air doesn't like to stretch.)
Eventually the droplet leaves the moist air that it is in. If it leaves out of the sides of the cloud, or the bottom, it may re-evaporate. On the other hand if it leaves out of the top of the cloud, which usually requires some pretty considerable convective wind transport, then it may get to the upper atmosphere where it can spontaneously freeze into “diamond dust”, tiny little flecks of supercooled ice. These are cirrus clouds, and they look a lot more fuzzy and flowy. A cumulus cloud that does this with those strong convective currents is a cumulonimbus cloud, a rain cloud. They have a characteristic anvil-like shape from the diamond dust top spreading out while the bottom is so firm and solid. (And the rain comes of course from the super-cooled ice crystals coming back down into the cloud, growing into big snowflakes, falling out of the cloud, and melting into raindrops.)
If you've been out on a foggy day then you have walked around in a cloud before, they do not have extremely sharp boundaries when you are up close in person. But, these very common cumulus clouds have this distinctive flat bottom and puffy top because those represent properties of the surrounding air where the droplets are appearing and disappearing. The bottom is flat because the dominant temperature and pressure gradient is vertical, the top is poofy with a sharp-looking boundary because you are looking at this boundary where this moist air mass from below collides with a dry air mass which it is pushing, roiling, bubbling into. The droplets at the one boundary are either turned away or evaporated into the dry air, the droplets at the other boundary are formed and are pulling more moist air up after them.
A: Clouds are fuzzier than they look.
Clouds get their white colour from Mie scattering of light from water droplets of size comparable to the wavelength of light. But for smaller droplets Rayleigh scattering is the best approximation. The formula for the intensity of that radiation seen by an observer at distance $R$, scattering angle $\theta$, wavelength $\lambda$ from a particle of refractive index $n$ with diameter $d$ is
$$
I=I_0 \left(\frac{1+\cos^2\theta}{2R^2}\right)\left(\frac{2\pi}{\lambda}\right)^4\left(\frac{n^2-1}{n^2+2}\right)^2\left(\frac{d}{2}\right)^6
$$
Note the last term: the scattering intensity increases with the sixth power of the diameter. That means that as the vapour density increases in the air and droplets start forming, even a completely smooth gradient of droplet sizes will look like it has a sharp edge where the scattering goes from minuscule to dominant.
Once droplets are $\sim 10$% of the light wavelength the full Mie theory is needed, but the effect is roughly the same (and less wavelength dependent).
There are doubtless other forces keeping cumulus clouds sharp, like the upper boundary often corresponding to the top of an upwelling convective flow into drier air and hence having a strong vapour gradient, and the cloud base being set by temperature, pressure and the dew point.
A: Convection in meteorology is often assumed to be more or less adiabatic, and convection is what creates cumulus clouds. You can imagine clouds as bubbles rising inside a water tank, they also have a quite well defined boundary. Warm humid air rises up, cools (reduction of pressure) and condensation begins.
https://en.wikipedia.org/wiki/Cloud#Adiabatic_cooling
https://en.wikipedia.org/wiki/Lapse_rate#Dry_adiabatic_lapse_rate
Furthermore the wind in the upper atmosphere is less turbulent than wind close to the ground. The clouds are advected by the general motion of the wind. So the warm humid air bubble which rose up stays together. With strong shear winds clouds some times get thorn apart.
To add to the question, there are cumulus clouds that look fuzzy as well. For example the Pileus cloud phenomenon.
https://en.wikipedia.org/wiki/Pileus_(meteorology)#/media/File:Sarychev_peak_eruption.jpg
Edit: To add to your question about periodic structures. There are periodic clouds. They are created by gravity waves in the upper atmosphere.
https://www.antarctica.gov.au/site/assets/files/51703/07-clouds_chris-wilson.1600x900.webp
