IANAL, but the legalese-to-Engligh translation is something like this.
There were four issues to decide:
1. Can the Supreme Court of the United States (SCOTUS) even decide this whole case now?
There is a law, the Anti-Injunction Act, that says “no suit for the purpose of restraining the assessment or collection of any tax shall be maintained in any court by any person.”
That means you cannot bring a lawsuit to prevent enforcement of a tax until after the tax has been collected (and, for practical purposes, you have asked for the money back. Since the penalty provision does not even begin until 2014, no one has yet paid it.
The court decided that the requirement that you pay a penalty to the IRS if you do not have health insurance is not a tax for the purposes of the Anti-Injunction Act. Therefore SCOTUS is ok to decide the other issues brought before it.
2. Is the individual mandate constitutional?
The Patient Protection and Affordable Care Act (variously known as PPACA or Obamacare) requires certain persons to have a health insurance policy or pay a penalty.
The government argued first that Congress has the right to implement the PPACA because the Constitution permits Congress to regulate interstate commerce and Congress can implement laws that are Necessary and Proper to carry out such things. The SCOTUS disagreed saying the regulation of commerce only extends to the regulation of existing activity, not inactivity. Further SCOTUS ruled that though it might be necessary, it was not proper for Congress to proceed in this manner.
The government alternatively argued that Congress has the right to levy taxes and the penalty for not having health insurance is such a tax. The SCOTUS agreed and thus upheld the individual mandate.
Yes, this means the government on one hand argued that this is not a tax and then turned right around to say it is. Legalese is funny that way.
3. If the individual mandate is unconstitutional then is it just that part of the law that is unconstitutional or should the whole law be overturned?
Since the individual mandate was ruled constitutional under the taxing authority of Congress this point was really no longer relevant.
4. Is the expansion of Medicaid under the PPACA constitutional?
Medicaid is a program in which the federal government and the various state governments use tax dollars to provide for health insurance for certain persons, mostly low income persons.
In the past the federal government has used incentives or threats to pull funding in order to get states to do various things that the federal government might not have the right to do on its own. The federal government threatened to pull a portion of federal highway funding from any state that did not set a certain maximum speed limit. A national 21 year old drinking age was forced through similarly.
The PPACA threatened to pull all federal dollars for Medicaid from any state that did not expand the criteria for who would qualify for Medicaid. The SCOTUS ruled this was unconstitutional because the funding in question was so large that it left the states with no realistic alternative than to comply.