A wig is one of those old fashioned fake hair methods where you just put it on top of your real hair with a monofilament top in one section of the wig, usually in the top.
A lacefront is where they put lace around your skin (the perimeters) and the scalp is more realistic. Usually, you can part them anywhere.
A half wig is where your hair is out and you use clips to attach it to the back 2/3 or 3/4 part of your own hair that is not out.