@redhouse, being independent means the government doesn't have power over it, like the courts, you don't want to make judges subject to the executive branch because then the executive branch has too much power...
In the case of the Federal reserve (or the European Central Bank) the basic principle is, we do not want the government to have the power to cause inflation... because someone might come along and decide that it's a good idea to win votes.
Being a private entity makes sense, you become a legal entity by virtue of the laws which make companies entities, this means that you have a legal responsibility which is spelt out in law. Any organisation of people can organise to become a company (whether they are a charitable organisation, an educational organisation, a profit-making organisation, a co-operative of farmers, or a fan club) This gives them certain protections in law - so the individuals can't be sued for doing task for the organisation.
While the organisation can be held responsible for it's actions - though I don't know how you can sue the federal reserve (for, let's say, screwing up the economy) - The European Central Bank has a single objective, it is ONLY allowed to maintain inflation levels at a low positive amount - so no deflation, no hyper-inflation - and it has a number of tools to do this, such as setting interest rates for money it loans to banks. Or... hmm, no maybe that's the main one(you can probably print euros aswell). This affects the money supply indirectly by altering the interest banks will offer on loan or savings, which will change some people's choices to save money or invest that money. (depending on the likely percentage earned from investment, which is a gamble, but is basically the same as the amount of economic growth, IF you take an average and invest in every company at the same time)
So the ECB is not allowed to promote economic growth, it has a specific job to do, and it is only allowed to do it's job. This is Euro-area policy for how to manage an economy, governments can use taxation (taking money in or spending that money) to stifle or promote economic growth (you might want to stifle the purchase and use of petrol/oil because europe doesn't have any major sources of oil, so the balance of trade means money/value leaves the european economies every time we buy foreign oil... - thus most european countries have huge levys on the consumers of oil, (so for example, 54% of the price of petrol at a pump in ireland is tax, that is more than half of what you are paying - or 130% tax - see:http://blog.aaireland.ie/index.php/2012/12/04/motor-fuel-taxes-ireland/a-breakdown-of-irish-fuel-taxes-and-hikes-in-ireland )
Obviosuly this differs from america.