The term 'Trojan asteroid' has become somewhat ill-defined. Originally, Trojan asteroids were specifically the swarm of asteroids that trailed Jupiter, located at approximately 60° behind the planet in its orbit. They were named thus by Johann Palisa, the astronomer who first described their orbit, and who suggested that they all be named after heroes of the Trojan War. These asteroids all reside in the Jupiter-Sun L5 Lagrangian point, where the interaction between the sun and Jupiter creates a stable gravity field.
There is a corresponding group of asteroids that precede Jupiter in its orbit, at the L4 point1; these Palisa called the Greeks, after the Trojan's opponents. While it remains traditional to name the asteroids in these groups after heroes of the appropriate armed force2, it has become common to refer to all of these asteroids as Trojan asteroids; if clarification is needed, you may refer to the 'Greek camp' and the 'Trojan camp'.
Moreover, it has become common to call the L4 and L5 points in any Lagrangian system Trojan points (they are also called the 'triangular Lagrange points'), and any asteroids at these points, in any system, Trojan asteroids (or, occasionally, 'Lagrangian asteroids'). Neptune and Mars have been found to have Trojan asteroids, and in 2010 the first Earth Trojan asteroid (technically, a Greek asteroid) was discovered; 2010 TK7, at Sun–Earth L4.
Unless otherwise specified, Trojan asteroids do mean Jupiter's Trojans (of both camps). There are currently over 5000 identified Trojans escorting Jupiter, and theories vary widely on the their total number. The total number of Jupiter's Trojans with diameters larger than 1 km is estimated to be anywhere from ~35,000 to over 1 million3. Each Trojan swarm is spread out over about 26° of Jupiter's orbit, or about 2.4 AU (a total of 4.8 AU, or about 12% of Jupiter's orbit).
1. The other Lagrangian points in the Jupiter-Sol system (or any other system, for that matter) are unstable, and contain nothing of interest.
2. Most of them are not named, although hundreds are. The majority of Jupiter's identified Trojans/Greeks (~65%) are found at the L4 point (the Greek camp). Over 6000 have been labeled, although most of these are still known by their provisional designation.(Source). This number no doubt includes a number of comets, composed of ice and dirt rather than rock, that have been captured by Jupiter.
3. These estimate vary based on the assumed albedo of the asteroids and theories on how evenly they might be distributed. We are currently working with very limited information. The higher estimates would put the number of large Trojans as approximately equal to the number of asteroids in the asteroid belt.