Suppose you have a linear dynamic system. That is, the function that predicts the next state from the current state to the next is linear. Suppose also that the states in your system are not known precisely but have some uncertainty modeled by a (multivariate) normal distribution. Then the uncertainty in the state at the next step also has a normal distribution, because a linear transformation of a normal distribution remains normal. This is a very high-level description of the classic Kalman filter.
When the transition from one state to another is nonlinear, the probability distribution around future states is not normal. There are many variations on the Kalman filter that amount to various approximations to get around this core difficulty: extended Kalman filters, unscented Kalman filters, particle filters, etc. Here I’ll just discuss unscented Kalman filters and particle filters. This will be a very hand-wavy discussion, but it will give the basic ideas.
It’s easy to push discrete random points through a nonlinear transformation. Calculating the effect of a nonlinear transformation on continuous random variables is more work. The idea of an unscented Kalman filter is to create a normal distribution that approximates the result of a nonlinear transformation by seeing what happens to a few carefully chosen points. The idea of particle filtering is to randomly generate a cloud of points and push these points through the nonlinear transformation.
This may make it sound like (unscented) Kalman filters and particle filters are trivial to implement. Far from it. The basic framework is easy to understand, but getting good performance on a particular problem can take a lot of work.