All-weather tires are a recent innovation from the tire industry. They are essentially all-season tires but with a slightly more winter-focused rubber compound and tread pattern. In other words, there will be more sipes across the tread, and the rubber will be more pliable in the winter.
Naturally, going for a set of all-weather tires will give you better traction and a lot more confidence in the winter, especially when compared to a regular all-season tire. A winter/snow tire will still be a better option for harsh wintry conditions, but a set of all-weather tires will work well for most people.
Most all-weather tires sacrifice a bit of dry/wet traction to achieve higher snow traction, but some, like the Michelin CrossClimate 2, work outstandingly well across the board.