Traditionally roasting used radiant heat to cook meat. This would have taken place over an open flame, typically on a rotating spit. In modern times this method is now called Rotisserie. Modern roasting refers to dry heat cooking that takes place in an oven, the food is cooked by convection. Until the late 19th century this method was referred to as baking.
Baking is essentially the same as roasting in modern times. Baking most often refers specifically to the cooking of "baked goods" (breads, pastries, etc.). However, the terms baking and roasting are often used interchangeably (baked chicken, roast chicken). There doesn't seem to be a hard and fast rule here as to which term is used. For example, you roast asparagus, turkey, chicken, but bake lasagna, casseroles, and also chicken.
Braising uses a combination of moist and dry heat to cook the food. This is what occurs in a crock pot, or in dutch oven/stock pot in the oven. Your meat will be cooked in an ample amount of seasoned liquid such as wine and/or stock.
Broasting I had never heard of before. According to Wikipedia it's a trademarked method of pressurized deep frying. I doubt this is what you were referring to.