Where is the Monte Carlo method used in physics?
Monte Carlo is a particular numerical technique heavily used in Physics, mainly when one needs to "brute force" the calculation.
Virtually, all areas of Physics can make use of simulations that include some sort of Monte Carlo code, from the designing of detectors in particle physics (think Fermilab and LHC: Geant4), passing by out-of-equilibrium statistical mechanics (search for meta-stable stationary states), all the way to Lattice QCD and Climate Modeling!
In fact, the techniques used to simulate Quantum Field Theories are heavily based in Monte Carlo and are flexible enough to be of value in several other branches of Physics (from Statistical Mechanics to Climate Modeling).
You can think in the following terms: in Physics, many problems are integro-differential in nature, i.e., they can be expressed either as a Differential equation or as an Integral one. Solving Diff Eqs can be tricky (think in terms of variable mesh sizes, etc: hyperbolic and parabolic diff eqs can make use of quite different mesh choices) and, sometimes, it pays to cast the problem as an Integral eq and use Monte Carlo to brute force the calculation of the answer, given that MC can be a very effective integration tool.
Nowadays, the basic tool-belt of a computational physicist includes Monte Carlo and LAPACK (ATLAS, BLAS) in a couple of different languages (C/C++, FORTRAN, python, etc): these tools, combined with an effective modeling and coding of the problem, can be as sharp as the state-of-the-art requires (think parallel processing, GPU, CUDA, etc).
In statistical mechanics, the Monte Carlo method is used to sample the state space of a statistical system in order to compute physical quantities of interest by averaging.
For example, a Markov Chain Monte Carlo (MCMC) method was used in the WMAP  papers, the most important measurment/papers of modern CMB  cosmology, one of the pillars of the Big Bang Theory of the Universe. MCMC was used to explore and fit the parameters of the theory with the WMAP (and other) measurements. You can download the technical papers at:
 Wilkinson Microwave Anisotropy Probe
 Cosmic Microwave Background
Another important use of Monte-Carlo is in nuclear engineering: basically for two distinct things: nuclear reactor core calculations and radiation shielding.
The first use Monte-Carlo to obtain precise estimations of the reactivity of the core (and other parameters) and the second allow to calculate, for example, the thickness of a shielding.
More generally this is related to particle-matter interaction : this is a process where each interaction is quantified by its cross-section which then gives a probability for the interaction to occur. This is thus a very good field to apply MC methods.
Another example is in medical physics, for the calculation of the deposited dose on a patient.