Family based immigration has been a core foundation of US immigration law and policy. There are basically two kinds of family-based immigrant visas: 1. Immediate Relatives of US Citizens (including spouses, minor children, and parents of US citizens who are 21 or over); and 2. everybody else (specifically, spouses and children of Lawful Permanent Residents, adult children of US citizens, and siblings of US citizens). Immediate Relatives of US citizen visas are available without limit; Family Preference visas are allocated based on criteria linked to the perceived priority of the need. You can learn more about family visas in the Immigrant Visa page and the related links.
For specific information about immigration for spouses and fiances, please read the article Love American Style.
To understand what the financial responsibilities are for family immigrants, look at What's a Public Charge
For specific information about Adoptions and US immigration, please read the article Adoptions