There really is no short answer to what a classical education is. It is something that was taught many years ago in all schools but fell out of fashion as education trends changed. It is a way of learning that families choose to do primarily as homeschool, though there are some classical education schools around.

Classical education has three stages: the grammar stage, the dialectic stage, and the rhetoric stage.  They are founded on the principle that at different ages you learn in different ways. How you teach a teenager is different to how you teach a 10 year old.

WHAT ARE THE 3 STAGES?

The grammar stage is the foundation of learning. It involves a lot of repetition. Children in the grammar stage are younger than 12 years old. It is a time period when kids naturally acquire & remember knowledge, facts, information. The instruction is more on learning facts. They do not need to know the hows and whys at this stage. Concrete ideas are memorized, not abstract ideas. Repetition is key in the grammar stage,

The dialectic stage is the stage from 5th or 6th grade to 8th or 9th. This is where your child starts to apply logic to what they have been learning. You introduce logic, critical thinking and long discussions into their education. You build on the foundations learned in the grammar stage. Scientific method is taught, ramifications of war are discussed, and you are teaching them to argue logically and with reason.

This leads to the third stage, the rhetoric stage. This is essentially 10th to 12th grades where you teach your teen how to analyze and synthesize their own thoughts, beliefs and opinions. It is the stage at which children are taught how to teach themselves anything. 

PROS AND CONS OF CLASSICAL EDUCATION

As with everything, there are pros and cons. Of course, how you look at them will have something to do with your educational bias. Not everyone likes classical education, or Montessori, or unschooling, etc. I am going to try and present an unbiased list of pros and cons to help you narrow your thoughts and ideas on whether this could be good for your family.

Pros

• Strong foundation in literature, esp the classics
• Appreciation for the arts
• As you work through the levels it produces good thinkers
• It follows the natural stages of development in children
• Children learn facts with the view that later they will view them logically

Cons

• Emphasis on memorization, narration and dictation. Not all children enjoy this. 
• Can be very parent intensive
• Emphasis on ancient languages
• It can be rigid. Therefore it doesn’t allow for exploration of topics you enjoy if you are following a full classical education curriculum.
• There can be deficiencies in math and science

Classical schooling works for many families. It suits their educational goals and aims. It might be a good curriculum for you. As with everything, it is ok to try something and then change it if it is not working for you.

Here are some other educational philosophies I have looked at:
Montessori
Waldorf-Steiner
Charlotte Mason

Some friends who follow classical education, like this book to help understand what it is all about: The Well-Educated Mind by Susan Wise Bauer