There are many ways to go about doing foster care. Some states and counties do them through the state level. Other areas use private agencies.
Once you contact one of these companies, you will fill out a large application that will include very detailed information on your life, from how you met your spouse to what was your greatest loss. Then they will do a homestudy of your house where they will check everything from how hot your water heater gets to where you keep your medications.
After you pass those, you will state what age/gender/ethnicity you are willing to accept through foster care. Then it is a waiting game. Once a child is in need of a foster family, they will give you a call with a very brief, sometimes inaccurate history of the child. You then state whether you are able to take them right then. The child will be placed with you, within hours of that call, and could stay with you for a night or they could become a permanent part of the family. Each case is different and no matter how certain your social worker is, they often cannot predict how long they will be staying.