Applying vacuum at the end of a flash chromatography column
All chromatography needs to work is some pressure differential, no matter how it's established. Gravitational chromatography uses hydrostatic pressure from the weight of the solvent, and a typical gravitational column can have a roughly ~20-50 mbar pressure difference between the top and bottom of the column; flash chromatography also has the hydrostatic pressure, but has an additional positive gas pressure at the top from e.g. pumped air, and can typically reach ~100 mbar pressure differences.

The principle of chromatography is to force compounds in a mixture to stay at the interface of a stationary phase (e.g. silica) and a mobile phase. The slight differences in how molecules behave in this boundary are added up over time, creating the observed separation.

Since this is a surface process, having a stationary phase with higher specific surface area and better packing effectively means molecules spend more time undergoing separation at the interface than being dissolved in the bulk of the eluent, where no separation happens. However, a more tightly packed stationary phase also restricts eluent flow. Under a simplified set of assumptions, decreasing the stationary phase particle size by half (and therefore increasing specific surface area by a factor of 4) requires 4 times the pressure differential to produce the same flow rate. So getting better separation in reasonable timeframes requires larger pressure differentials.

This is where vacuum chromatography techniques come in. Vacuum pumps are a staple laboratory instrument, and can easily achieve negative pressures of ~900 mbar with common glassware. Therefore, without too much hassle, it is possible to obtain good flow rates using finer stationary phase particle sizes and get improved separation. 

Of course, larger pressure differentials can be achieved with positive pressure at the top of the column. Medium, high and ultra-high pressure chromatography (MPLC, HPLC, UHPLC) make use of stationary phases with very fine particles (from ~10 μm to ~1 μm diameter) which start to look more like silt or clay. The only way to force liquids through these stationary phases is with very large pressure differentials, from ~5 bar to as much as ~1000 bar, but they can achieve incredible resolution of compounds during separation. The reason UHPLC isn't used for every single separation is cost and scalability; it's an engineering challenge to control such enormous pressures safely. Glass is very resilient under compression (which is why glassware can commonly hold a vacuum), but weak under expansion. This means such positive pressures need much more rigid containers, such as thick steel columns. This also limits their feasible sizes and therefore the amount of stationary phase/eluent/compound they can handle.

As an example of a vacuum chromatography technique, I point towards the excellent video by Daniel Pedersen detailing dry column vacuum chromatography (DCVC). There is also a preceding publication which goes into further details. For what it's worth, I've found this vacuum chromatography technique to be far superior to flash chromatography.