Te best nonpartisan analysis is the CBO report on the first version (pdf). Some details change, but the general idea remains the same.
For the first ten years Romney-Ryan (1) converts Medicaid into a block grant program, with much lower funding than projected under current law and (2) sharp cuts in top tax rates and corporate taxes. Note this doesn't reduce the deficit, it’s a tradeoff of reduced aid to the poor for reduced taxes on the rich with much higher tax reductions than savings.
How do they claim a big deficit reduction? The answer is two big “magic asterisks”. The first is Congress, with Republican control, will increase taxes in a broadening but unspecified way. Remember, these are GOP Congresspeople who never vote for tax increases and sign pledges to never do so but somehow they will. Expert opinion is that this will never happen especially with the magnitude of the tax increases this plan needs. The second "magic asterisk" is discretionary spending being cut in half as a percentage of GDP with details not specified. No proposals are specified on the massive cuts to popular programs needed.
After ten years Medicare is transformed into a mainly private insurance plan scheme with the government providing money, in the first plan vouchers, for people to buy this insurance. However, this second phase doesn't count on Medicare savings for deficit reduction but instead on further unspecified cuts in government programs that were already reduced 50% in the first ten years. Discretionary spending, which includes the military currently at 4% of GDP, is supposed to fall to 3.75% from the current 12.5%.
Nobel prize winning economist Paul Krugman summarizes:
Ryan basically proposes three big things: slashing Medicaid, cutting taxes on corporations and high-income people, and replacing Medicare with a drastically less well funded voucher system. These concrete proposals would, taken together, actually increase the deficit for the first decade and beyond.
All the claims of major deficit reduction therefore rest on the magic asterisks. In that sense, this isn’t even a plan, it’s just a set of assertions.
Putting some numbers on this fantasy wish list - tax cuts for millionaires and corporations $4.3 trillion. Off set that with cutting Medicaid $0.8 trillion, add somewhat less than $1.0 trillion in cuts to education, food stamps, job training, everything else, and lastly in this first version cut $0.7 trillion in Medicare. That's it, there isn't nothing more except for the "magic asterisks" that say the Republicans in the House will raise taxes and find more cuts. So $2.6 trillion INCREASE in the deficit in the first ten years as no one expects the Republicans to pull the magic unicorns out of their asses and raise taxes and cut the military and cut all programs even more than the $1.0 trillion already assumed.
For the second version of the plan released in 2012 replace vouchers for Medicare with another form of payments to individuals to pay for private insurance that are just like vouchers, but not vouchers because people don't like vouchers, instead it is called premium support. The savings from Medicare come from the government payments being less than what the plans cost and that gap increases each year. That is the basis of Democratic advertisements, you pay more for Medicare under the Romney endorsed Ryan plan and that quickly mounts to many thousands of dollars each year.
Ryan budget two moved balancing the budget back to 2040, 28 years of deficits!, because no one liked the cuts proposed. There were some specifics - all deductions and credits on the income tax form eliminated, hope you didn't need any of them, in favor of somewhat lower rates. Reduce the top corporate rate to 25% from 35%, part of why corporate giving of hundreds of millions has swung to Romney. The military budget is held constant with inflation, there goes the only real spot to cut hundreds of billions but it does preserve the Republican campaign donation stream. Ryan 2 reduces discretionary spending 2% in 2013 from prior budget by another cut by fiat with no details. Overall, except for what I just mentioned, more vagueness in cuts and taxes. Here is a chart of the Republican fantasy federal budget.
The Romney plan based on the Ryan plan expanded the vagueness but set five tax goals and added sweeteners for billionaire and big business, must keep those contributions coming. The five goals dooming the mathematics of the Romney tax plan are: 1. Tax reform must be revenue neutral but 2.) must reduce income rates by 20% and 3.) must preserve or increase deductions for investments. 4.) Eliminate the Estate Tax. 5.) Get rid of the Alternate Minimum Tax.
A problem: just using those assumptions independent analysts that Romney had praised in getting the nomination found the plan would greatly reduce tax rates for millionaires and increase income taxes for everyone else. The Republicans cried foul for injecting facts into the debate. After consulting with the Romney camp and other conservative Republicans the analysts issued another report sticking by their conclusions: voting for Romney will increase your taxes for the great majority of people. Even ending the tax exmption for government bonds and insurance investments, violating plank 3 but a Republican suggestion, and even allowing for increased growth because of the magic fairies who love small government, your taxes will go up.
After being hammered by the media and Democrats about the Medicare cuts Romney flip-flopped last Sunday to say he would not cut Medicare by $0.7 trillion. That increases the deficit under the Romney plan by $0.7 trillion to $3.3 trillion in the first ten years. I should point out that Obama's $0.7 trillion in Medicare SAVINGS expanded and provided more Medicare and health care benefits and also that overall ObamaCare reduced the deficit according to the CBO but that the same savings in the Ryan plan did not expand health care. Now the repeal of Ryan's cuts by Romney just adds to the deficit. Romney says he will use the $0.7 trillion to extend Medicare's life until it is fixed. But Romney has flipped-flopped on so many issues, in this case Medicare in just a few days, so often can anyone trust him? Originally Romney-Ryan plans just took the $0.7 trillion out of Medicare causing it to run out of money in 2016 so this flip improves that at the cost of the budget deficit.
The fact that Romney's plan is mathematically impossible is not the worst problem. The fact that Republicans repeatedly and routinely lie about what they will do (it just came out that the GOP has issued a directive to candidates to not use the words "entitlement reform" or "privatization") is not the worst problem. The problem is that Republicans will say and do anything to get elected and then sprinkle magic pixie dust on their policies claiming one thing while actually doing what they have done for decades - reduce taxes on the rich and hang everything else. Romney is just a more obvious example of that.
The real Romney-Ryan tax and budget plan is simple - reduce Romney's taxes and hang everything else. That is the plan, all the other words and goals and statements about the deficit and everything else don't matter - the plan is to cut taxes for the 1%.