The "Walstad" setup is termed after Diana Walstad, who in her book, Ecology of the Planted Aquarium, spoke of establishing an aquarium ecosystem where the plants and fish balance each others' needs. Here, the plants act as the water purifier, as opposed to the mechanical filter. Rather than converting ammonia to nitrites to nitrates, plants can directly convert ammonia from fish waste to plant mass. Plants also remove chemicals and heavy metals that may be present in the tank. The fish provides CO2 to the plants through respiration, and the plants growth in response provides oxygen for the fish.
Following this natural approach, the same can be accomplished for a smaller container, such as a vase or a bowl. A smaller container serves as the perfect introduction to new aquarists, or for terrestrial gardeners looking to get their toes wet.
Glass bowls and vases range from a half gallon to 10+. If you are planning to keep fish -- the larger, the better -- go for a 10 or 20 gallon aquarium tank. If you want to get by with smaller, a 2 gallon minimum is required for a betta fish, but a 5 gallon+ is recommended. A larger body of water offers greater stability, and more room for error. For smaller containers (1-2 gallon), you can make do with a couple snails or dwarf shrimp, but keeping a fish in one would just cause you endless headaches.
Dirt + sand is not only the most economical choice, but also very effective. I use approximately 1" of Miracle Grow Organic Garden Soil ($10 @ Home Depot), not because it's organic, but because there are no perilite or too many larger wooden chunks that float to the top. Sift through some of the large pieces, and lay it down into the vase. Cover the top with 1" of play sand ($5 for 50lbs @ Home Depot), this keeps the soil to the bottom, and your water clear.
Add water slowly, I recommend covering the sand with a plastic bag and pouring water against the side of the container. You may still end up upsetting some soil into the water, just net out the floating pieces, and wait for the rest to settle.
Plant a larger variety of species, generally cheaper plants are easier to grow. Organize your plants from smallest to largest as foreground / midground / and background. A few recommendations:
Foreground - dwarf baby tears (med-high light), marsilea minuta (all light), blyxa japonica, amazon microsword
Midground - limnophila mini, green camboba, various crypts, lindernia parviflora
Background - ludwigia arcuata (easy red plant), star repens, limnophila aromatica, rotala macranda
Floaters - red root floaters, dwarf water lettuce, salvinia minima, frogbits
The fun part is experimenting with different kinds of plants :)
A fluorescent desk light (2W/gallon, lower if LED or if receiving sunlight), i.e. this tiny IKEA LED grows all the plants in the 2g bowl in the first picture above. Recommend 10-12 hour cycles on a mechanical timer.
Do not add shrimp or fish until the plants are well established. I would recommend waiting a month. Snails on the other hand can go in on day 1, I recommend getting a few malaysian trumpet snails, these move through the substrate, aerate them, and distributes waste materials into the substrate, to be liquefied by nematodes, then sucked up by the plant roots.
Though not recommended, if you insist on adding fish to a very small vase (<5g), consider a small filter or power head to help cycle the bioload.
6. Maintenance and Heating
The vase or bowl requires very little maintenance, besides water top offs due to evaporation, and plant clippings due to overgrowth. The soil should sustain the plants for the first 6 months or so. Afterwards, if the plants begin to droop, insert osmocote tablets into the substrate as needed (every 2 months).
If you live in colder climates you will need a heater for the tank. Most common fish and plants are kept in a temperature range between 70-80F, most hobbyists keep their heaters turned to approximately 78F. You need 4W of heating per gallon of tank water, it's best to place the heater near the water output of your filter or pump so warmer water circulates evenly into the aquarium.
7. Algae Control
You may encounter algae problems sometime down the line, especially on the glass. On a square container it's easy to clean with a magnetic algae cleaner, but not on a round vase or bowl. Instead, buy 1 nerite snail online or from your local fish store. These little guys constantly move up and down the glass and eat most kinds of algae. For long term control, plant more, more plants means less nutrients for the algae, thus less algae, add in some floating plants to suck up excess nutrients.