[Bootstrapping][1] is a resampling method to estimate the sampling distribution of your regression coefficients and therefore calculate the standard errors/confidence intervals of your regression coefficients.

 1. The nonparametric bootstrap resamples repeatedly draws your
    observations **with replacement** (i.e. some observations are drawn only once, others multiple times and some never at all), then calculates the logistic
    regression and stores the coefficients. This is repeated $n$ times. So you'll end up with 10'000 different regression coefficients. These 10'000 coefficients can then be used to calculate their confidence itnervals.
    To really have stable estimates, I would suggest more than 1000
    replications, maybe 10'000. You could run the bootstrap several
    times and see if the estimates change much whether you do 1000 or
    10'000 replications. If your bootstrap estimates vary between your estimates and the observed, single model, this could indicate that the observed model does not appropriately reflect the structure of your sample. The function `boot` in `R`, for example, puts out the "bias" which is the difference between the regression coefficients of your single model and the mean of the bootstrap samples.
 2. When performing the bootstrap, you are not interested in a single bootstrap sample, but in the distribution of statistics (e.g. regression coefficients) over the, say, 10'000 bootstrap samples.
 3. I'd say 10'000 is better than 1000. With modern Computers, this shouldn't pose a problem.
 4. What do you mean "the results vary each time"? When your number of replications is high enough, the bootstrap should yield very similar confidence intervals and point estimates every time.


   [1]: http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-bootstrapping.pdf