Two epidemic threshold parameters are derived for the spread of a highly infectious disease in a community of households, where a household is any group whose members have frequent contacts with each other. It is assumed that the infection of any member of a household results in the infection of all susceptible members of that household. The threshold parameters have simple expressions in terms of the mean household size and the mean and variance of the number of susceptibles per household. They provide a basic reproduction number R0 for the spread of infection from individual to individual and a basic reproduction number RH0 for the spread of infection from household to household. The threshold parameters are used to derive the levels of immunity required for the prevention of major epidemics in the community. They are also used to evaluate various vaccination strategies having the same vaccination coverage. For a community with households of equal size, it is found that random vaccination of individuals is better than immunizing all members of a corresponding fraction of households. In contrast, when households have varying sizes, immunizing all members of large households can be better than a corresponding vaccination coverage of randomly selected individuals. It is illustrated that these threshold parameters can also be used for a community of households with schools or day care centers. In particular, the effectiveness of immunizing all members of a school is quantified.