This is a very well-known part of the wedding vows. It means "until we die". This is supposed to mean that people who marry each other are going to stay married for their entire lives.
People sometimes use this phrase when they're talking about how long a marriage is supposed to last:
The rate of divorce continues to rise in this country, and that's really disappointing for those of us who still believe in "'till death do you part".
Are women who are being abused supposed to just stay in the relationship and put up with it until death do them part?