I am currently marking essays from some of my Open University students about sources of English Law. Arising from this there are a couple of issues about bills in parliament which it would be useful to clarify.
Bills are either (1) Public - affecting public interests. Most of the bills now considered are public - ranging from the Mental Health Bill to the Legal Services Bill. (2) Private - which relate to private - individual or legal individual (a corporation or local authority) - interests.
Public bills may be introduced by Ministers (Government Bills). These have most chance of success because, save for specific time set aside, Government business takes precedence. Ordinary members (backbenchers) can introduce 'Private Members Bills'. They are public bills!
Most bills are introduced first to the House of Commons. My current reckoning is that this session, for the introduction of public bills, the figures are
Government Bills - Commons -19; Lords - 9
Private Members Bills - Commons - 31; Lords - 21
Ten Minute Rule Bills - Commons - 20
The ten minute rule bill procedure (Standing Order 23) is a device to allow a backbencher to raise an issue of importance in 'prime time', though the "bill" he asks leave to bring in is almost never proceeded with.
There are some excellent factsheets available at http://www.parliament.uk/parliamentary_publications_and_archives/factsheets.cfm