This "thing" is called a surround. It could be foam, it could be rubber, or it could be paper.
If it's not torn and the glue is just separated, you can glue it back. Buy speaker surround glue from Parts Express.
If it's foam and the foam is torn, you can buy a replacement foam kit from Parts Express also. (I've just done a set of car speakers and the kit worked great). Just buy the kit for the correct size speakers and follow the directions enclosed.
If the surround is torn but it's rubber or paper, glue probably wouldn't fix the tear correctly and the speaker should be reconed IMHO.
