For simplicity we are viewing finite 2-dimensional surfaces "floating" in 3-dimensional space.
Some, such as a closed disc or an multi-holed anulus, have boundaries and some, such as a sphere or multi-holed tori, do not. A torus has a "donut hole" which is intuitively obvious to see but very difficult to define. The thing is, a hole is can not be viewed locally "from" the surface itself, but must be viewed as how the surface itself is "shaped" in space.
A "hole" is a 3-dimensional hole in the space the surface occupies. If you think of the 2-dimensional surface as the "skin" of a 3-dimensional solid, the solid has a hole in it but the skin is intact.
A "cuff" is a 2-dimensional hole that is in the surface itself. If you cut a hole out of the skin of a sphere and the insides can now leak out.
The subtle part is that a cuff need not appear as a hole in the skin, it could be the border of a surface. Imagine a sphere. Cut a little patch of it and you have a giant bubble with a hole in it. But stretch the hole out and you have a bowl with a rim. Flatten it out and you simply have a closed disc. The cuff has become a border.
Consider a sphere. Cut two cuffs in it. Stretch those cuffs apart so that that remaining surface becomes cylinderical. Then you have a straw. Shrink one cuff and widen the other and flatten and you have an anulus.
Interestingly if you glue the two cuffs of a straw together you get a torus.