It involves a water proof container (most probably plastic) with a cover, the nutrient solution is filled in the container and the plants are anchored on the container's cover such that their roots are barely touching the water. The plant roots grow and absorb water with nutrients, the plant grows, blooms and fruits... depending on the plant type and container size, no additional nutrients or water is required till the season lasts... e.g. Lettuce can easily grow in a 5 litre bucket but tomato and cucumber require 70 litre to complete their life-cycle. On the other hand small buckets can be very carefully re-filled on weekly-monthly basis..
I tried my hand on growing tomatoes in a 16 litre used paint bucket.

Used this Hole-saw to drill holes for plants in the cover:


Used mineral water bottle was cut in half, each half will be used to anchor plants on the plastic lid/cover of the container:

Two plants which were growing in soil were taken out and put in the bottles, i placed some coco-chips to cover the exposed roots, these pics are taken after two weeks of placing the plants in the Kratky setup:

Roots are already grown in the water:

The formula I am using contains the following:
N (NO3-) 170 PPM
N (NH4+) 19 PPM
K 310 PPM
P 41 PPM
Mg 45 PPM
Ca 150 PPM
S 112 PPM
Fe 5 PPM
So far so good, the setup is on my roof top for maximum sunlight.