Homeschool teachers do have some leeway in regards to curriculum choice, hours and days of operation, and even the name of their school, but the United States does still have certain rules that each homeschool must follow in order to be legal.
Laws do vary for each state, so be sure to check out your particular state before you open your homeschool. There may be rules regarding national testing, number of days you must operate, and whether or not you have to implement a national test at the end of each year. While the laws may seem a little intimidating at first, most are simple, easy to implement, and things you would do anyway just for organizational purposes and record keeping.
There are several websites available for you to find the homeschool laws for your state. Just do a simple google search for "homeschool laws". To be more specific you could include your state, (i.e. "homeschool laws in North Carolina". The following are a few of the websites I have found that give very detailed information regarding homeschool laws in each state.
As always, do not rely on others to give you the legal information you need. Be sure to do your own research. There are literally thousands of websites out there that give you very detailed information regarding your state's homeschooling laws. As an extra precaution, always cross-reference each site to ensure that they are providing the same information.