Socialism: general term for the political and economic theory that advocates a system of collective or government ownership and management of the means of production and distribution of goods.
This is not what Hillary wanted to do, nor what Dean did in Vermont. In a Socialistic environment, everybody contributes to the central government, and then everybody gets the same level of service. Now of course, that's not to say that wealthy people can't go outside of the system to get better healthcare (at their own expense of course), but everybody gets the same basic level of service.
With Dean/Hillary care, you and I would get no basic level of service for being a member of The Great Society. You and I would have to continue paying our insurance premiums, plus pay extra to cover the costs of everybody who can't afford insurance. There's no provision for us to say "I don't want to pay $600 a month for medical insurance even though I'm making $100,000 a year, so just give me the basic US Healthcare service." That's not the way socialized medicine works, but it is a great example of a "rob the rich to buy votes from the poor" scheme.