A Monte Carlo method is a method of carrying out mathematical calculations and related simulations that uses chance to estimate the answer. Calculations are carried out using numbers chosen at random, yielding an estimate of the solution. It is named for the city of Monte Carlo, which is famous for gambling, something that also depends upon chance.
For example, to find the area of a weird shape for which there is no obvious direct calculation, if there is an easy way to determine whether a specific point is within the weird shape, take random points throughout a rectangle that circumscribes it, then count out what percentage of these points fall within the weird shape. That percentage of the area of the rectangle (an area that is easy to calculate) gives you an estimate of the area of weird shape. Choosing the random points must be done in such a way that they are randomly distributed over the rectangle. The more such points, the better the estimate.
Quite a few challenging math problems are "solvable" by such chance methods. Markov chain Monte Carlo (MCMC) is a more specific type of method that incorporates chance. Quantum Monte Carlo methods are the use of Monte Carlo methods to solve quantum system problems, e.g., using quantum mechanics.