Is it the idea that ALL women living today have this particular mitochondrial DNA?
No - but all people alive have inhereted it from their mother. Like other DNA, it is not the same for all people. As nwr briefly explained above, the relative differences can be used to determine approximately how distantly related two people are.
Men DO have this mitochondrial DNA, right, they just don't pass it on, right?
Yup, the sperm isn't big enough to carry it - but the egg is.