How does the Higgs boson generate the masses for all other particles? Is it the carrier of a force?
You need to distinguish between the Higgs boson and the Higgs field. The Higgs field is the stuff that gives all other particles a mass. Every particle in our universe "swims" through this Higgs field. Through this interaction every particle gets its mass. Different particles interact with the Higgs field with different strengths, hence some particles are heavier (have a larger mass) than others. (Some particles have no mass. They don't interact with the Higgs field; they don't feel the field.) It is the opposite of people swimming in water. As people float in water they "become" lighter. Depending on size, shape, etc, some people float better than others.
The Higgs field is not considered a force. It cannot accelerate particles, it doesn't transfer energy. However, it interacts universally with all particles (except the massless ones), providing their masses.
The Higgs boson is a particle. It gets its mass like all other particles: by interacting with ("swimming in") the Higgs field. But as you can imagine, the Higgs particle differs from all the other particles we know. It can be thought of a dense spot in the Higgs field, which can travel like any other particle. Like a drop of water in water vapor.
The Higgs boson has many more ways of interacting with all other kinds of particles than the Higgs field (which just causes a "drag" = mass). In this sense one might call the Higgs particle the mediating particle of the proposed Higgs field, like you wrote. The Higgs field is the silent field that gives the mass. We cannot directly probe for it. But discovering the Higgs boson, the "mediator", would prove the existence of the Higgs field.
The Higgs particle, like many other elementary particles, is not a stable particle. Since it interacts with all kinds of other massive particles it can be created in collisions. (The Higgs particle does not interact with massless particles, such as a photon or a gluon. Since these particles don't interact with the Higgs field, the Higgs boson also doesn't interact with them.)
Once the Higgs particle has been created, it will eventually decay. Though the Higgs particle interacts with all massive particles it prefers to interact with the heaviest elementary particles we know, especially the top quark, which was discovered at Fermilab in 1995. Because of this property of the Higgs boson physicists at Fermilab might have a chance to find evidence for the Higgs boson itself within the next five to six years. If they are not successful then an accelerator currently build at the CERN laboratory in Geneva, Switzerland, will have enough energy to produce the Higgs boson. Fermilab's accelerator currently is the world's most powerful accelerator, but physicists don't know whether it has enough power to create Higgs bosons. The new accelerator at CERN will have more power, but construction won't be finished until 2005.