If you have registered with a local government or business as domestic partners, ending that registration probably requires nothing more than filling out a form. If you are married and currently live in a state that recognizes same-sex marriage, follow your resident state's divorce procedures. Likewise, if you still live in the state in which you registered as domestic partners, follow your state's dissolution procedures -- most of these states treat domestic partnership dissolutions like divorces and require partners to follow divorce procedures in family court.
However, if you got married or entered a domestic partnership or civil union in a marriage-equivalent state and you currently live in a state that does not recognize the relationship, ending the relationship can get very complicated. Most states have a residency requirement in order to use the divorce courts. So you can't just travel to the state you got married in and file for divorce. Similar issues arise with domestic partnerships. Your best bet is to contact a divorce attorney who is well-versed in same-sex couple breakups and get advice on your specific situation. To learn more, see Nolo's article Breakup Advice for Gay Couples Who Married or Registered in More Than One Place.
For a practical guide, updated with the latest legal information and legislation, that will help you and your same-sex partner protect and exercise your rights, see A Legal Guide for Lesbian & Gay Couples, by Denis Clifford, Frederick Hertz and Emily Doskow (Nolo).