A model is a set of assumptions that describes relationships among observed characteristics, data, values, and events.
As highlighted by Emanuel Derman, models in finance can have, at least, three different meanings: a fundamental model, which is a system of postulates and data, together with a means of drawing dynamical inferences from them, a phenomenological model, that is a description or analogy to allow the visualization of something that cannot be observed and a statistical model that is a regression or best-fit between different data sets.
In regulatory frameworks such as SR11-7, the term “model” refers to a quantitative method, system, or approach that uses statistical, economic, financial, or mathematical theories, techniques, and assumptions to process input data into quantitative estimates. A model is formed by three components: the information input component, the processing component and the reporting component.
The difference between a model and an algorithm is that a model is a set of assumptions while an algorithm is a recipe to solve a particular model. As one example, the Hull-White model can be used to describe the behavior of the interest rate curve. The underlying assumptions are a.o. that the short rate is driven by a mean reverting stochastic process. To compute the NPV of a derivative as given by the Hull-White model, we can use a Monte Carlo algorithm to perform the numerical integration.