Using household vinegar as weed killer
Vinegar, when applied to weeds, can be an effective weed killer, but it's important to understand the mechanism. Household vinegar is a weak acidic solution (5% acidic acid). Acids will strip off the protective coating on plant foliage; leading to desiccation of the leaves, killing that portion of the plant. The root system is not affected, so in most cases the plant will recover and need retreatment, especially if there is a tap-root, like dandelions, which extends into the soil. Full strength vinegar is necessary to have any effect, and adding a small amount of dish soap (1 oz/gallon) will improve effectiveness. Treatment should be done on a warm, dry day of at least 65° F. This article, Vinegar Weed Killer, has more information.
There are more concentrated vinegar (20% acidic acid) weed killers available. These are more effective, but as stronger acidic solutions must be handled with care (gloves, protective clothing, skin and face protection). It's important to just apply the solution to the weeds. If you spray the soil area enough you can change the pH, and disrupt the soil structure and microbes that are the basis for soil fertility.
This article from OSU, Profiles of Natural Pesticides, discusses the use of natural of natural pesticides including using vinegar as a herbicide.