This is not exhaustive, rather a good start on differing educational philosophies. I recommend learning your philosophy before purchasing curriculum or signing up for a homeschool program.
Biblical/Christian
This approach is based upon Biblical reasoning and a Biblical or Christian worldview, which requires considering and pondering the purpose of everything in God's universe. The curriculum is designed for parents and educators who recognize their children as gifts from God. Therefore, their solemn, God-given responsibility is to educate their children with a program that meets their physical, social, and spiritual needs. This philosophy often follows the traditional approach, and the curriculum is complete and inclusive as described under Traditional. Some curriculum leaders in this philosophy are Abeka, Bob Jones, Sonlight, Calvert, Catholic Heritage Curricula, Kolbe Academy, Konos, My Father's World, Oak Meadow, Rod & Staff, Seton Home Study School.
Charlotte Mason
A method of education in which children are taught as whole persons through a wide range of exciting living books, firsthand experiences, and good habits. Mason's philosophy of education is probably best summarized by the principles given at the beginning of each book in her six-volume set. Two critical mottos from those principles are "Education is an atmosphere, a discipline, a life" and "Education is the science of relations." She believed that children were born persons and should be respected; they should also be taught the Way of the Will and the Way of Reason. Her motto for students was "I am, I can, I ought, I will." A Charlotte Mason, or CM, educational philosophy features short lessons (10 to 20 minutes per subject for younger children and longer for older ones), emphasizing excellent execution and focused attention. Narration, 'living'' literature, nature study, artist/picture study, and habit training are some of the unique aspects of a CM education. Websites that you might find helpful are: Simply Charlotte Mason and Ambleside Online.
Classical
The heart of a classical education lies in the trivium - the grammar, logic, and rhetoric stages. These coincide with natural learning abilities during childhood. For most families, classical education includes a dedication to in-depth studies of Latin, mathematics, the arts and sciences, and a deep understanding of world history. Many classical homeschoolers follow a four-year or six-year cycle of repeating science and history topics. Some helpful websites might be A Well-Trained Mind, Classical Conversations, or Sequitur.
Eclectic
Eclectic homeschooling is choosing to teach using many different forms - for example: using the Trivium, using living books, having your children narrate to you what they just read, doing a unit study when you begin learning about Ancient Egypt or Light and Color and having lots of engaging materials and manipulatives of all kinds in your home for every subject.
A traditional definition of "eclectic" from the American Heritage Dictionary says, "1) Selecting or employing individual elements from a variety of sources, systems, or styles: an eclectic taste in music; an eclectic approach to managing the economy. 2) Made up of or combining elements from various sources: 'a popular bar patronized by an eclectic collection of artists, writers, secretaries and aging soldiers on reserve duty."
As for curriculum, Eclectic homeschoolers will use resources precisely as they see fit for their family and not necessarily as the instructions indicate. In this way, the Eclectic homeschooler uses many different homeschooling philosophies to accomplish their educational goals. Absolutely EVERYTHING can be considered a curriculum. For more information, visit EHO or Young Scholars.
Montessori
Although this method is often used in a school setting, it is experiencing growing popularity among homeschooling families. Montessori education is an educational approach developed by Italian physician and educator Maria Montessori. Montessori education is characterized by an emphasis on independence, freedom within limits, respect for a child's natural psychological development, and technological advancements in society. Although a range of practices exists under the name "Montessori", the Association Montessori International (AMI) and the American Montessori Society (AMS) cite these elements as essential: Mixed-age classrooms with classrooms for children aged 2½ or 3 to 6 years old by far the most common; Student choice of activity from within a prescribed range of options; Uninterrupted blocks of work time; A Constructivist or "discovery" model, where students learn concepts from working with materials, rather than by direct instruction; Specialized educational materials developed by Montessori and her collaborators. For more information on using this method at home, visit the International Montessori Index of Schools. Here's Montessori Homeschool Curriculum.
Traditional
This method most closely follows a traditional school model and strives to mirror that type of classroom setting in the home. The study proceeds according to written lesson plans. Daily work is turned in and graded, lessons are followed by tests, and grades and records are kept. A report card may be issued on a quarterly or semester basis. Some families relax this approach somewhat, still following structured schedules and grade levels, but choosing their own curriculum and creating their own lesson plans. The traditional method looks very similar to how you were likely taught in school and is sometimes referred to as school-at-home, structured homeschooling, scope and sequence schooling, or school-in-a-box. Traditional homeschoolers usually purchase a complete curriculum, which includes textbooks, parent guides, tests, schedules, and grading and record-keeping materials.
This method follows a structured schedule each Monday through Friday, September through June, following a traditional school system. Oftentimes this curriculum is available on DVD, CD-ROM, or the Internet. Some people choose to integrate some traditional subjects with another philosophy. Although there are many curriculum sources for this philosophy, here are a few places to start looking - Clonlara School Home Based Education Program or Oak Meadow. Many of the curricula under the Biblical heading offer Traditional homeschooling curricula too.
Some curriculum leaders in this philosophy are Abeka~Bob Jones~Sonlight~Calvert~Catholic Heritage Curricula~Kolbe Academy~Konos~ My Father's World~Oak Meadow~Rod & Staff~Seton Home Study School.
Unit Studies
The basic concept for a unit studies homeschooling approach is using one topic to incorporate most of the school subjects. Learning occurs around a central topic, a subject, or an area of learning. One popular site for a unit studies curriculum is Konos. Unit studies are a popular homeschooling method because they are typically hands-on, literature-based, and can be molded to fit into other homeschooling philosophies. Individual topics can encompass all of the scholastic subjects or be very specific to only cover science, history, or a literature selection, for instance. Unit studies can be found on the Internet for almost any subject you or your children can desire. Simply type "(topic) unit study" into your Internet search engine.
Unschooling
Unschooling is also known as "natural learning", "experience-based learning", or "independent learning". Many children coming from other learning environments or school settings spend some period of time "deschooling" before becoming ready to unschool. Unschoolers believe that children are sufficiently motivated to educate themselves by their own natural curiosity. The student chooses how, when, why, and what they learn. The parents act as facilitators and advisors rather than teachers. Emphasis is on providing an environment rich with opportunities (google "strewing") for experimenting and exploring, along with a wealth of resources and support. Some additional websites for more information are Sandra Dodd and John Holt.
Waldorf
Waldorf education is based on the ideas of Rudolf Steiner, an Austrian philosopher. Learning is interdisciplinary, integrating practical, artistic, and conceptual elements. The approach emphasizes the role of the imagination in learning and developing thinking that includes a creative and an analytic component. The educational philosophy's overarching goals are to provide young people the basis for developing into free, morally responsible, and integrated individuals and to help every child fulfill his or her unique destiny. It can appear similar to Unschooling and Montessori approaches. A Waldorf education requires more description than is possible in this space. Please visit these websites for a more complete picture of this educational philosophy: Waldorf Without Walls, Christopherus Homeschool, or Waldorf Homeschoolers. Here's a website that offers a free preschool curriculum.