A Dutch oven is a large, heavy kettle or pot used for slow cooking, especially outdoors over a fire. The cookware is usually made from cast iron, and has a tight-fitting lid that keeps heat in. Dutch ovens are specialty cookware, and tend to take up a lot of space. However, you don't have to own one to cook recipes meant for a Dutch oven.
The glass or stoneware casserole dish usually comes with a tight-fitting cover, and can replicate the steamy environment of a Dutch oven. Unlike a Dutch oven, casseroles don't allow for frying or sauteeing before roasting. Perform these operations in a separate pan, then place food into the casserole and cook in a conventional oven at the temperature and time called for by your recipe.
Slow cookers resemble electric versions of Dutch ovens, and can be used for many of the same operations. The stoneware crocks hold heat in a similar way to the Dutch oven's cast iron sides. Some slow cookers allow for sauteeing inside the crock. Slow cookers work at a lower temperature and take longer to cook than a Dutch oven on a stovetop. Adjust recipes accordingly.
Clay pots are traditional cooking vessels all over the world. These unglazed earthenware pots soak up water, creating a moist, enclosed environment similar to that inside a Dutch Oven. They don't work well directly on a stove top, so use a heat diffuser between the pot and the heat source to prevent cracking.