Under the same circumstances, a third magnitude comet is usually harder to see because the same amount of light is distributed over the area of a fuzzy blob. The scale changes logarithmically. A star of first magnitude is about 2.5 times brighter than a star of second magnitude, which is 2.5 times brigher than third magnitude, and so on. As mentioned above, each star actually has two magnitudes. For appareant magnitude, the measurement is taken exactly as the star appears.
Absolute magnitude is measured differently.
Around the second century BC, the stars seen in the sky were divided into six brightnesses by Hipparchus - a Greek astronomer. He called the brightest stars first magnitude, and the dimmest stars sixth magnitude. Much more recently around the 1850's, this scale was formalised by Norman Podgson of Radcliffe Observatory. He made it so a first magnitude star was 100 times brighter than the faintest star which could be seen without a telescope. The biggest limit with magnitude is that it only measures light we can see.