Gluten is a protein that's found in wheat, cereals, and grains. Many people these days suffer an immune reaction to gluten, commonly known as coeliac disease, where they experience abdominal problems from consuming gluten. This prevents many people from drinking beer, which is brewed with grains. Since wine, however, is made from grapes and contains less than 20 parts per million of gluten, it is generally considered gluten free.
There are a couple instances in which gluten can come in contact with wine such as ageing and fining. It is very uncommon these days, but some wine makers use a flour based paste to seal the wine barrels which can contaminate the wine during ageing. The process which filters out unwanted particles from the wine to make it clear is called fining. Gelatin and isinglass are the most commonly used agents during the fining process but this causes issues to those who want vegetarian or vegan wine. To create vegetarian or vegan wine, gluten can be used instead, but this causes issues for coeliac sufferers. After the fining process, centrifuge can be used to remove the gluten but this is costly and the amount of gluten present is usually very minuscule.
The good news for those effected by coeliac disease is that it's pretty safe to consume all the delicious wine they want! Any wine that's produced with higher than the legal requirement in the UK and the US (20 parts per million gluten) would have to alert buyers on the bottle's label.