Regulations vary by state, but to teach in the public school system, it's likely that you'll need a teacher certification. This might entail passing a subject examination for the subject you wish to teach, along with showing that you have experience within the classroom, among other requirements, like fingerprinting and a background check. You'll need to have a college degree, but it doesn't have to be in education. For example, if you wish to teach chemistry or French, majoring in those fields will probably help you more than majoring in education. Check your state's Department of Education for the exact requirements for your area.
If you weren't planning on being a teacher, you probably haven't taken any courses in pedagogy, nor had any experience as a student teacher. Understanding classroom management is an important part of your success in the classroom. A teacher certification prep program can help fill in the gaps when it comes to teaching and help you earn your state's certification.
Private schools aren't always as regulated as public schools. Some will hire people who don't have certification to teach, as long as the person can show competency in the classroom. In these cases, the subject of your degree is likely to have an effect on whether they'll hire you. In the elementary years, a school is probably looking for someone with a background in early childhood education. However, at the high school or junior high level, you may be able to secure a position that relates to what you studied in college.
Though it's not required to have have a degree in education in order to get your foot in the door, earning a master's or Ph.D. in education can definitely help you get further in your career. In many cases, teachers who hold advanced degrees are paid more than those who don't. Advanced degrees can also help you move into the educational administration fields, if that's where your interest lies.