Sirius is the brightest star in the night sky, here astronomers actually meant Sirius has the highest apparent magnitude (about -1.5 magnitude) in the night sky and the two terms: apparent magnitude and luminosity are totally different in some ways. Apparent magnitude is a measure of a star’s brightness or any astronomical object from a particular point or place in space whereas luminosity is the intrinsic brightness of a star, in other words, how bright the star really is. Astronomers usually compare the brightness of two objects using the luminosity of the Sun, which is about 4 x 10^26 W. Other than that, luminosity of a star does not change with the observer’s distance to the star, that means, two observers situated at different distance from the star should get the same reading of luminosity of the star no matter where they are.
Luminosity is the intrinsic brightness of a star, in other words, how bright the star really is.
Dependent on surface temperature
A high luminosity star like Canopus in the constellation Carina does not necessarily mean it has a high apparent magnitude too. Being extremely distant from Earth, its apparent magnitude will be affected, it will be lower in this particular case. Here we can conclude that luminosity is not proportional to the apparent magnitude of a star if the distance between the star and the observer is not constant. Furthermore, the brightness of the star is dependent on surface temperature and radius measure. For example, a star with an identical temperature with our Sun, but with a bigger radius or bigger size, will have a higher brightness than our Sun. Similarly, a star with the same radius as our Sun, but with higher surface temperature, will have a higher brightness than our Sun.
We can calculate the brightness of a star using the formula L=R^2 x T^4, where L is luminosity, R is radius and T is surface temperature. Here we normally use ‘solar’ as our unit for radius and temperature. 5 solar radii, for instance, means 5 times the radius of the Sun (695,510km). The formula can be targeted to other objects in our sky such as interesting comets and asteroids.