Cornstarch (called corn flour in the UK) is made by grinding part of the corn kernel into a powder.1 This white powder can be used to thicken soups, 
sauces, pie fillings, etc. Cornstarch can be added directly to cold liquids, but to avoid lumps cornstarch must be mixed with a cold liquid (usually water or stock) before it can be added to hot liquids like soup or gravy. This mixture of cornstarch in a cold liquid is called a slurry.
Cornstarch, potato starch, arrowroot powder, and tapioca powder are all comparable in terms of thickening ability, so you can usually substitute them 1:1. There are some differences, however, as some thickeners are less suitable when combined with certain ingredients (e.g. arrowroot does not work well with dairy and cornstarch is less effective in acidic liquids).2
On the other hand, flour is only half as effective as cornstarch, so if you are using flour instead of cornstarch or one of the others named, you'll need to use twice as much.
