What does it mean if country X has a higher GDP at Purchasing Power Parity (PPP) than country Y? I understand the implications of having a higher/lower GDP at market exchange rates. I also understand what PPP measures. 

But, what does a difference in GDP, at PPP, indicate in terms of differences in levels?
I like this explanation from Callen (2007), published in the IMF's F&amp;D, their quarterly magazine.  For measuring consumption in a country, PPP is probably the way to go, but international economic and financial power is probably better measured with market rates.

PPP versus market rates
So which method is better? The appropriate way to aggregate economic
data across countries depends on the issue being considered. Market
exchange rates are the logical choice when financial flows are
involved. For example, the current account balance represents a flow
of financial resources across countries. It is appropriate to use the
market exchange rate to convert these flows into dollars when
aggregating across regions or calculating the global current account
discrepancy. But for other variables, the decision is less clear cut.
Take real GDP growth. International organizations use different
approaches. The World Bank uses market-based rates to determine the
weights in its regional and global aggregations of real GDP, whereas
the IMF and the Organization for Economic Cooperation and Development
use weights based on PPP rates (although the IMF also publishes a
global growth aggregate based on market rates in the WEO). Each
methodology has its advantages and disadvantages.
Advantages of PPP
A main one is that PPP exchange rates are relatively stable over time. By contrast, market rates are more
volatile, and using them could produce quite large swings in aggregate
measures of growth even when growth rates in individual countries are
stable. Another drawback of market-based rates is that they are
relevant only for internationally traded goods. Nontraded goods and
services tend to be cheaper in low-income than in high-income
countries. A haircut in New York is more expensive than in Lima; the
price of a taxi ride of the same distance is higher in Paris than in
Tunis; and a ticket to a cricket game costs more in London than in
Lahore. Indeed, because wages tend to be lower in poorer countries,
and services are often relatively labor intensive, the price of a
haircut in Lima is likely to be cheaper than in New York even when the
cost of making tradable goods, such as machinery, is the same in both
countries. Any analysis that fails to take into account these
differences in the prices of nontraded goods across countries will
underestimate the purchasing power of consumers in emerging market and
developing countries and, consequently, their overall welfare. For
this reason, PPP is generally regarded as a better measure of overall
well-being.
Drawbacks of PPP
The biggest one is that PPP is harder to measure than market-based rates. The ICP is a huge statistical undertaking,
and new price comparisons are available only at infrequent intervals.
Methodological questions have also been raised about earlier surveys.
In between survey dates, the PPP rates have to be estimated, which can
introduce inaccuracies into the measurement. Also, the ICP does not
cover all countries, which means that data for missing countries have
to be estimated.