Is sampling from data, getting a statistic and repeating this process to get a mean of the statistic a "Monte Carlo" method?  Suppose we have some data (say around 10,000 cases) that sample a variable of interest; e.g., the concentration of metal in soil.  By generating a random series of indices we choose some of those samples and then compute some statistics, for instance the mean. We do this many times and finally get the mean of mean values.  


*

*Is this the Monte Carlo method?  

*If yes, how? If not, why not? What is needed to be a MC calculation?

 A: What you describe is "a" Monte Carlo method (but there is not just one MC method so it is not "the" MC method).
Monte Carlo methods refer to a large group of methods that involve repeted random sampling.  Exactly what you are doing depends on the sizes of your resamples and whether you are sampling with replacement or not.  If your resamples are the same size as the original sample and you sample with replacement then this is called "bootstrapping".  If your resamples are without replacement and smaller than the original sample then it is a variation on the "jacknife".  There are many other methods that qualify as Monte Carlo methods as well.
A: I'd say calling this a Monte Carlo method is a bit misleading.
I think that what you do is usually called Simple Random Sampling, see here:
http://en.wikipedia.org/wiki/Simple_random_sample
and is particularly used in survey analysis, polls, etc.
Monte Carlo methods usually refer to sampling methods used in integral approximations, where you choose the number N of draws, and the larger N, the more precise your results are. They are usually only justified asymptotically, when N goes to infinity (in which case they are expected to give the "right solution", usually to compute an integral of interest). In this respect your method seems different from classical MC methods, as introduced e.g. here:
http://books.google.com/books/about/Monte_Carlo_statistical_methods.html?id=HfhGAxn5GugC
A: This sounds like a bootstrap or at least a variation on the bootstrap. If you are resampling from the full dataset you have access to, and importantly the resample size has the same n as the original, this would be a bootstrap. I would not label this a monte carlo. I think of monte carlo as when you are actually running simulations on fake data, often to test a particular method, to see if you get the answer you should expect given the parameters you specified in the fake data setup. 
