Usually nothing, unless the deceased partner made a will or used another estate planning device, such as a living trust or joint tenancy agreement. In a few states, same-sex partners who are registered as domestic partners or have entered into a civil union relationshipmay automatically inherit a portion of a deceased partner's property, but these laws are by no means the safest or easiest way to plan for inheritance. The bottom line is simple: To protect the person you live with, you should specifically leave property using a will, living trust, or other legal document.
For help making a property agreement, as well as answers to more questions like these, see Nolo's Living Together: A Legal Guide for Unmarried Couples, by Ralph Warner, Toni Ihara, and Frederick Hertz. The book includes various agreements as tear-out forms and on CD-ROM, and provides state-by-state charts of state laws for your reference.