A match safe is any container that keeps matches safe from heat, damp, or prying children. They range from the elegant vesta cases carried by the gentlemen of the Victorian era to heavy-duty canisters for campers to tin boxes to hang on a kitchen wall.
These days, matches are not used by most people on a regular basis, and the most common type of match safe is simply a heavy-duty, watertight canister about the size of a salt shaker, used by hikers and other adventurers to keep their matches dry. These cases may be made of metal or plastic, and often have a loop on the end so that it can be attached to a lanyard, and may also include unrelated features such as a compass set into the cap. In the grand tradition of the modern outdoors-man, these are often ridiculously over-designed, made from aircraft grade aluminum and having redundant strikers, in addition to the different striking surfaces for strike-anywhere matches and safety and storm proof matches.
More traditionally, a number of designs exist for house-hold match safes. These exist primarily to keep the matches handy to light candles and stoves, and may be a decorative box, a knickknack of infinite tacky designs, or a integral part of a candlestick. However, by far the most common type of house-hold match safe is a small tin or wrought iron box that can be attached to the wall. In the American South, the most common form is a box taller than it is wide, (~13 x 8 x 4 cm) with a open feeder 'scoop' at the bottom (two pictures). Also common is a box of the same general dimensions, but with the long thin side against the wall, and the wide sides becoming the top and bottom; the top was usually lidded (picture). These types of match safes are still in use for those of us with wood stoves in our homes.
The most collectible type of match safe is that used by smokers in the days before butane lighters. These were small cases, usually slightly smaller than a pack of cards, and often quite fancy. It was common for them to be made of silver, although they could be made from anything from tin to ivory. They were often heavily decorated, and designed to show off one's style and wealth. Because one of the more popular brands of matches used by smokers was Swan Vesta, these are most often known as vesta cases. These might have various fancy features, such as cigar cutters and knife blades. It was also common for these to have an attachment for a chain fob, and these were often used to help justify the use of a Double Albert vest chain.
Given that the heyday of match usage was only about 100 years (~1830 to 1930), the enormous range of match safes is rather astounding. A Google search for "Match Safe" finds tens of thousands of different designs.