This is a simple question that actually has many complex answers depending on the context in which it's asked. This is also one of the most controversial issues in Indian affairs today. Anthropologists, ethnologists, lawyers, government agencies, different groups of Indians and non-Indians all have their own ideas about how this question should be answered.
American Indians or Native Americans are the people that were indigenous to North and South America. In general, Indians are the descendants of these people. In addition, there are several specific definitions of "Indian".
A federally recognized Indian, sometimes known as a "card-carrying Indian" is a member of a federally recognized tribe. Federally recognized tribes each use their own criteria for determining who can be a member of that tribe, and these criteria can vary widely from one tribe to the next. Some tribes require a certain "blood quantum" or degree of Indian "blood", others require only that a person have a direct ancestor on a certain roll or census of the tribe taken at a particular time.
Federally recognized Indian tribes are tribes that the federal government acknowledges it has a special nation-to-nation relationship with, usually because of treaties or agreements between the two nations. The federal government has certain obligations to these tribes and tribal members. However, the fact that a tribe or individual person is not "federally recognized" does not necessarily mean that they aren't Indian, at least as far as the federal government is concerned. The United States recognizes that there are Indian tribes with which it doesn't have special obligations, and it also recognizes that there are individual Indian people who aren't members of a tribe.
State recognized tribes are eligible for some federal programs. Indians who are not members of a tribe can be eligible for certain federal programs if they can establish that they have an Indian cultural identity, and having Indian ancestry does not automatically mean that you have such a cultural identity. For the purposes of the U.S. Census, individuals can declare their race as whatever they consider themselves to be.
States can recognize Indian tribes according to the state's own definition. The federal government recognizes these state recognized groups as Indian, it just does not have any special relationship with them or obligations to them.
Traditionally, Indian communities defined their members according to an individual's cultural identity rather than genetic background. A person who is born and raised in a community will obviously share the cultural identity of that community. Many Indian societies also had procedures for inducting outsiders into the community. These procedures usually involved, at least in part, the outsider adopting the culture of the community. Today, there is much debate among the members of some Indian tribes and communities about how they define their members.
To sum up, Indians are the descendants of the original people of North and South America. Indian tribes decide who their members are. Not belonging to a tribe or having non-Indian ancestry does not necessarily mean a person is not an Indian. Government programs can have different definitions of the term "Indian", depending on the program. Indians usually have some degree of Indian "blood", but Indian communities, whether "recognized" by the government or not, decide who their members are.