A distribution is a distribution regardless of whether it was calculated by Monte-Carlo methods, obtained from an empirical sample. or assumed from some analytic distribution.
For distributions like the Normal distribution, the intervals are symmetric but it doesn't always have to be this way. You can for example a chi-square distribution which is skewed and your region (confidence interval) will not have the mean smack bang in the middle but closer to the lower value.
Basically the idea is that for some given alpha value, the chance of getting the true value in a particular interval is 100(1-alpha)%. So if alpha = 0.05 then the chance of the true parameter lying in the interval that was generated is 1-alpha where alpha lies in [0,1] and is typically 0.05.
In other words if we constructed millions of intervals from a sample with alpha 0.05 then we would expect that 95% of the time the true value would be in that interval.
This idea is the basis of fundamental statistical theory.