The improved Euler method (also called the midpoint Euler method or second order Runge Kutta method) attempts to use the average of the rate of change at the beginning and end of a timestep to reduce numerical error in a numerical integration of a differential equation.
Interpolation refers to the process of estimating an intermediate value from two know values. You might assume that if you had a full tank of gas on Sunday, and a half tank of gas on the following Saturday, that if you drove more or less the same every day that you probably had about 3/4 of a tank on Wednesday.
The process of creating numbers that simulate randomness on a computer is known as pseudorandom number generation. The "pseudo" in pseudo random refers to the fact that if you use a rule to generate a number, it is by definition not random, though it may appear so, and be close enough to random for all practical purposes.
Monte Carlo modeling refers to the solution of mathematical problems with the use of random numbers. This can include both function integration and the modeling of stochastic phenomena using random processes.