""The British system seems simpler. The House of Commons seems to be the only house that matters (I don't really understand what the House of Lords does nowadays.)""

The House of Lords, used to be occupied by hereditary Peers of the realm (hence the name), and was reformed by the Blair government to remove most of the Lords and replace them with New Labour appointed Peers.

It is our upper chamber of government, and its purpose, now and in the past, is to balance the potential excesses of the Government of the day.

All government legislation is examined by the Lords, and amendments suggested, where they feel that said legislation is badly drafted. They will also, on very rare occasions, actually refuse to pass a piece of legislation which they consider ill advised in toto.

Ultimately, the government can override the Lords by use of a device called the Parliament Act, but usually it is persuaded to make changes, to avoid delay in enacting new legislation.

I guess you would call it a safety valve which protects the public from bad Law, though bad 'uns still slip through (Public Entertainment Licencing 2003).

Hope this helps.
Don T.