I am obtaining confidence intervals using different bootstrap methods. When comparing them, what is does it mean to use Monte Carlo variation to look at how much the upper and lower limits vary for each bootstrap?
The confidence intervals that you estimate with a bootstrap will be subject to uncertainty.
There are different ways that one might estimate said uncertainty
One option is to do so analytically. Alternatively, one could use a Monte Carlo simulation to estimate the uncertainty.
Choice of methods probably depends on the type of bootstrap being used.
For example, analytic techniques seem a lot easier for a parametric residual bootstrap than a non-parametric residuals bootstrap or a paired bootstrap.
Richard is right. For example it is not just that the intervals vary with sample size but they are only asymptotically accurate. Also the higher order bootstraps such as bootstrap t, BCa and double bootstrap are usually more accurate than the lower order bootstraps such as the percentile method becaause they converge to their asymptotic confidence levels faster. But for some parameters such as for confidence intervals for variances even for moderate sample sizes the higher order bootstraps undercover and can for certain sample sizes even be less accurate than the lower order bootstraps. This can be determined by applying Monte Carlo to approximately calculate their coverage. This was shown by Chernick and LaBudde (2010) in the American journal of Mathematical and Management Science.