Volunters in the garden can be of two types. In a big garden you can have volunteers to help with work and they can be vital for maintenance. The other type is a plant that grows of its own accord without being planted. There is a thin line between a weed and a volunteer and I tend to only use the word to refer to potatoes, but I guess it can be used more widely.
In the case of potatoes, volunteers are those shoots that appear in the ground the year after potatoes were planted in a plot of land. They are undesirable for a number of reasons. Firstly they grow into big plants that may overwhelm the crop that year. Secondly they may be infected with blight and provide an early source of infection in your garden, potentially affecting tomatoes and potatoes. Potatoes are also affected by other pests and diseases and these volunteers can maintain these problems from one year to another even though you buy ‘clean’ seed potatoes every year.
It is why it is important to carefully dig up your spuds and avoid leaving small tubers in the ground. If volunteers do appear you should pull up the shoots or, even better, spray them with a systemic weedkiller so the roots are killed too. The photo above shows some land that has two different volunteers: lots of horseradish (Armoracia rusticana) that I spent years trying to get rid of, even though it was originally planted (not by me) as a crop – but is very difficult to get rid of – and a few potatoes from the previous year that I hoped might help swamp the horseradish – see I can be a foolish optimist at times. The next year I hoped that squash might help shade out the horseradish but they took a while to get going, by which time the horseradish had already made good growth.
Make sure you dig over the ground well when you grow spuds, to remove even small tubers.