The genus Fragaria includes all the strawberries, the most familiar of which are hybrid cultivars selected to give large, sweet, and flavorful fruit. However, there are a number of species of strawberries that have not been bred and still live wild. In fact, you may have some in your yard and not notice them, as they may not stand out from the rest of the forbs living in the shady margins of tree-shaded areas.
In the English-speaking world, 'wild strawberries' most often refers to Fragaria vesca, a variety that has spread throughout most of the Northern hemisphere, and is most commonly found in North America and Europe. Fragaria vesca is also known variously as woodland strawberry, alpine strawberry, Carpathian Strawberry, European strawberry, or fraisier des bois, depending on location. In any case, this is a small plant with runners sending out stalks terminating in three broad, serrated leaves, small, white flowers, and occasionally small red fruit. The fruit is edible, although neither very sweet or flavorful, being a cross between bland and bitter, with just a touch of strawberry flavor. It is sometimes harvested and eaten, although it is likely that any varieties that are harvested commercially have been bred to be more flavorful.
Another common wild strawberry is Fragaria viridis, which ranges throughout Europe and central Asia. They are reportedly quite pleasingly tart, although do not taste much like other strawberries. The fruit does not always redden when it ripens, and so in Europe these may be called green strawberries or creamy strawberries, while in Russia these are called Klubnika.
Also in Europe, Fragaria moschata, aka the musk strawberry or hautboy strawberry (a corruption of the French name hautbois), is said to have exceptionally good taste and scent, and is used in gourmet cooking. They are said to have the flavor of raspberry or pineapple (this is a common comparison for strawberries, as the most common commercial strawberry is sometimes called the 'pineapple strawberry'), and they are dark maroon in color. If you see varieties such as Capron or Profumata di Tortona, these are musk strawberries. These are hard to transport and store, but easy to grow.
The Japanese strawberries Fragaria yezoensis and Fragaria iinumae have spread to Southern Russia, presumably primarily imported as a decorative plant. They can be eaten (and young Fragaria iinumae plants are sometimes used in soups), but they are, like most wild strawberries, not very flavorful.
Other species show a fairly narrow variation from this range. While all strawberry species have seven chromosome, they may have different polyploidy, many having the standard two sets of these chromosomes, but others having anywhere from three to ten sets (scientists have managed 32-ploidy in hybrids created in the lab, but 10 seems to be the most occurring in wild strawberries). In general, more chromosomes means bigger organs, including the fruit.
Wild strawberries tend to grow in a wide range of habitats, tolerate poor soils well (although they prefer acidic soil), do well in shade, and in fact prefer growing under trees. However, they do not do well with temperature variations, and some varieties can be aggressive and take over your garden, so research before you plant a border. Keep your eyes open, there's a good chance you have some wild strawberries creeping around your neighborhood, if you can find them.