Lime juice is the juice of any one of a number of cultivars of the lime, a greenish fruit with a sour pulp and acidic juice. There are many different types of lime, with the most common being:
Citrus latifolia, also known as the Persian lime, the Tahiti lime, and the Bearss lime (after John Bearss, who developed it in 1895). These are the most common supermarket limes, and are seedless.
Citrus aurantifolia, also known as the Key lime or Bartender's lime. These are smaller and more yellow than the Persian lime, with many seeds and are not only highly acidic, but also interestingly bitter at the same time.
Citrus hystrix, also known as the Kaffir lime, or Makrut lime. These are about as small as the Key Lime, but have a warty skin and are intensely sour. The kaffir lime (both zest and especially leaves) are commonly used in Malaysian and Thai cooking, but the juice is more commonly used as a cleanser.
All lime juice is sour, having a pH of between 1.8 and 2, which makes it more sour than lemon juice. For cocktail usage, the lime's strong green, grassy aspect works well with herbal flavors (think gin and Green Chartreuse). It is not as neutral of an acidifier as lemon juice.