Don't get overwhelmed. Yes, there are many approaches but after you decide on one, you can always change later after you get used to homeschool. I've changed 3 times and am about to change again! I am a relaxed/eclectic homeschooler and am about to change to the unit study "Learning Adventures".
School At Home: This style is what a lot of people think of when they think of homeschooling. You may have a "school room" with desks and maybe a chalkboard. Mom teaches "class" and "school" is a very organized structured event. This style uses the textbook approach.
Popular publishers for "school at home" would be Abeka, Alpha Omega, BJU & Christian Liberty.
Unschooling: This approach is the opposite of school at home. Some unschoolers allow their children to learn through natural life experiences. Some unschoolers have a "child led" or "interest led" approach. A child would be the leader in their education and pick what they want to learn about and how they want to learn it. Generally, there aren't workbooks, textbooks or schedules.
Unit Studies: This approach is based on one particular topic and all subjects are weaved into that topic. Example: study the Civil War, read books on the Civil War, write papers or essays on the CW, study history during the time of the CW, and do arts or crafts relating to that time period. Great examples of unit studies are Weaver, Learning Adventures, Heart of Wisdom and Five In A Row.
Charlotte Mason: This method focuses on respecting the child. Nature walks, living books, journaling, dictation, art, good habits and narration are some of the ideas in this method. You can read more in the book called A Charlotte Mason Education.
Classical Education: I'll be honest, I don't know much about this style. There are 3 levels to learning: the grammar stage (grades 1-6), the dialectic stage (grades 7-9), and the rhetoric stage (grades 10-12). These stages are called the "trivium". You can read more about classical education here. The Well Trained Mind is a popular resource for this style.