For multicomponent vesicles, the line tension of domain boundaries and the component-dependent elastic properties (e.g., spontaneous curvatures) are the two most important factors that mediate the budding behaviors of these vesicles. This paper especially focuses on their effects on the budding types and the budding number of a two-component vesicle. We found that the budding number is mainly determined by the component-dependent elastic properties while the budding types mediated by line tensions. A phase diagram is also obtained showing three different types of phase regions: (i) partially budding, (ii) partially bud-off, and (iii) totally bud-off regions. These inspiring results are, however, derived from a very simple spherical-cap model, and have been tested by computer simulations showing good agreement. We emphasize that, besides testing the spherical-cap model, the computer simulation techniques developed in the current work can be easily extended to other systems involving multicomponent vesicles.