There several approaches here; many private schools mirror the State system as regards structure and expectation, but some allow / promote a more autonomous style of education.
Parents pay standard termly fees which cover everything from premises to teacher's salaries and must also pay for uniforms and trips. Some scholarships may be available for families on lower incomes. By no means all families who send children to private schools are wealthy; many choose to spend income on fees rather than have a new car, bigger house or a foreign holiday.
Time investment from you, the parents
At least one of you will need to shape your day around dropping off at school and picking up, helping with homework, sometimes facilitating involvement in Saturday morning sports activities, extra- curricular mid-week activities and supervision in half-terms and holidays. You may also be asked to volunteer to help out at school and possibly to become involved with fund-raising activities at weekends. Both of you are likely to need to be in some kind of paid employment in order to afford the fees.
Having a say
Parents generally sign up for the school policies on important matters and are unlikely to have influence on such issues as time-tabling or staffing. However, schools are keen to keep pupils and are mindful of the fact that parents have a choice. For this reason, parents probably in most cases have more of a chance of being listened to by a head at a private school than they do in a mainstream school, if they have a specific matter to raise. They are also more likely to be able to take their child on term-time holidays.
What the children learn
Private schools are not obliged to follow the National Curriculum, but many do and in addition many offer a wide range of extra-curricular activities. The Steiner schools have their own specific curriculum ( http://www.steinerwaldorf.org/differences.html) and Montessori schools offer the children a wide range of detailed learning experiences and allow the individual child to tackle these challenges at their own pace and in their own order. (This last point is included for completeness; there are currently no Montessori schools for children over the age of 5 in Cambridgeshire, although there is a Montessori school for children 4 to11 in Bury St Edmunds; Arbor School).
The learning environment
In most private schools, children as in the mainstream system spend their time in groups of other children all of the same age. The difference here is that class sizes tend to be much smaller, typically 12 or 14 to a class, (as opposed to 24 to 30+ ) In most private schools, young children are generally required to remain seated at a table for most of the time, to follow an agenda set by the teacher and to move at the pace of the group. Children are required to change activity regularly during the day at the pace determined by the time-table and there is an expectation that children will learn a certain amount by a certain date in a certain way. They generally have little scope for spending time with older and younger children, moving around as they learn, choosing their activity or exploring a topic at their own pace and in their own way. Possible exceptions to this are Steiner and Montessori schools, some of which permit the child more choice and flexibility.
The object of the exercise
is generally as with mainstream, to deliver a curriculum; i.e. to present a certain number of topics in a certain amount of time and to test whether and to what extent the child has retained the knowledge, but probably, given the smaller class sizes and being primarily answerable directly to parents rather than to L.A.s, with a greater emphasis on and chance of meeting the individual child's needs.