We may not do justice to the topic without mathematical expressions but an attempt could be as follows. We need to introduce some concepts, and that we are dealing with time series of observations. Some of these concepts maybe be used synonymously depending on the literature but here are the definitions :
- Forecasting : It means we build a view of the world by a mathematical/physical model based on the historical observation to make predictions about future of the system we are modelling.
- Smoothing : It means we try to make historical observations as "clean" as possible, thinking of removing noise when we try to listen a person.
- Interpolation: We try to obtain information about missing observations using observations close by in time.
- Filtering/data assimilation: This is to use data to build a probabilistic view of the data via incorporating our model.
Kalman filter possibly can lead to smoothing, filtering and forecasting as a whole package. So in Kalman Filter, we incorporate data and model, usually discrete differential equations, with the probabilistic model of the data. In Kalman's case probabilistic view is normal distribution, i.e., multi-dimensional bell curve.