The residual and institutional models are two different approaches to addressing social welfare issues, such as poverty, hunger and health problems, with the government providing funding and social work. The residual model generally holds that the government should be involved in social welfare only as a last resort safety net when other avenues fail. The institutional model favors continuing intervention as needed, seeing government help as a natural and normal occurrence in people's lives.
The Residual Model
The residual model of social work and social welfare essentially sees government support for people's well-being as a safety net of last resort. When poor people are unable to help themselves through the market, usually by working, or get help from family, friends or other social ties, then and only then should the government step in with aid necessary to fit their needs.
The kinds of programs produced under this model are generally seen as being limited to the poor. Support is often cut off once people have the means to get assistance elsewhere. As a consequence, these kinds of programs can be cheaper to administer than other models and can fit people's notion of justice in that someone isn't receiving something for nothing, but they can also provide less support than programs produced under different models.
Programs for the poor such as food stamps, Medicaid and emergency housing assistance generally fit this model in the United States.
The Institutional Model
In this model, social welfare is provided for essentially everyone in a society, rich or poor, and is considered part of what the society should be providing for its citizens. Programs developed under this model often don't have the problems residual models programs do with people phasing in and out as they go up and down in wealth and prosperity, but they can be more expensive to administer and can draw the ire of people who favor a limited approach to government.
Public schools and libraries tend to fit in this model in the United States, since they're developed under the understanding that they'll be available to everyone from all walks of life regardless of need, and are seen as a normal aspect of life, rather than something available only in cases of dire need.
In some countries, medical care and post-secondary schooling, such as college and trade school, are also administered through such a model, while in the United States these services are generally only available to the needy through a more residual approach.