Smart growth refers to a way in which cities can most intelligently deal with population growth, new construction and the addition of new businesses, transportation networks, etc.
Cities can utilize the services of a City Planner, who is trained and experienced in dealing with such growth, and has resources to draw on and plans to address the needs of a city in that situation. They can bring in environmental consultants to address water quality, supply, treatment and access issues, as well as habitat preservation plans.
They can adopt a 25 or 50 year plan for growth, where they anticipate the issues associated with growth as opposed to merely responding to them. Then they can create sensible zoning laws that make intelligent use of the cities transportation and infrastructure, and protect residential neighborhoods that make the city attractive for investment and keep people living there long term.
My wife is a city planner of the sort that the first answer refers to. According to her, smart growth, at least in the context of the US, means building cities with higher population densities and less "sprawl."
If cities do this, then people will be closer to the places where they work, shop, play, etc. This means you need to have businesses, homes, parks, and industry all relatively close together. This will allow for more walking and biking instead of driving. Higher densities will allow for more transit because more people will live within walking distance of bus or rail lines. All of this will lead to a decreased dependence on fossil fuel.
So the core idea of smart growth is to use fewer resources. To do that, you need to keep cities relatively compact so that less driving is required. This means having businesses and even industry closer to people's homes and it means having more apartment buildings and fewer single family homes. This is one reason why it is so hard to get places to follow smart growth principles.
If you follow the first link below, you can find the US EPA's 10 principles of smart growth.