Kosher Meat: Kosher meats are a) kosher animals b) slaughtered according to Jewish law. Certain internal injuries can render that meat un-kosher. Because of the possibility of internal injuries or signs of illness (such as scarring), there is a post-mortem internal inspection of the animal. Not all injuries to the animal render it treif. And not all organs are always checked because the likelihood of injury is very small in many organs. The lungs are always checked.
Glatt Kosher Meat: "Glatt" is Yiddish for "smooth." Usually it refers to the lungs, and it implies that the lungs are free of injuries. However, Ashkenazim allow for the removal of some injuries without rendering the rest of the meat treif. Fowl, fish, and non-animals cannot be made "glatt" kosher. Any such product is a marketing ploy to convince you that the item is "super kosher."
Beit Yosef Meat: Beit Yosef meat (aka glatt Beit Yosef meat) is, in simplified terms, the Sephardi interpretation of glatt kosher. In effect, they have a no-tolerance policy for any of those internal injuries that the Ashkenazim would remove. My understanding is that this is the majority holding of the Sephardi world.
If properly inspected, "regular" kosher meat is kosher, at least for Ashkenazim. Many, if not most, Ashkenazim hold by glatt kosher instead. In fact, if a kosher certification organization certifies non-glatt meat, they will likely be called "an unreliable hashgacha/hechsher." (For more, see What Are Hechshers and Why Do I Care?) This is something to be aware of, particularly if you plan to host meals or bring food items to a potluck.