How can we describe decision tree in laymen's language and what are the major fields that require this?
Decision Trees are pretty straight forward to understand.
Take for example a famous problem where you have to label each passenger on the Titanic.
For each person you have a bunch of info (
age, for example), and the
outcome after the disaster, whether they lived or not.
DT, tries to find the best pattern in order to correctly classify a person.
This is almost like what you'd do, for example you may think (correctly) that women and children were the first to get into a safeboat, and so
sex = Female and
age < 18, would be two pretty good first splits of the data.
These are good because they let you discriminate well the overall observations (in
dead), because there's a good portion of subjects that are either
Children that survided.
DT does this, but with some kind of measure of how good a variable splits the data, the variable that discriminate most is the first, and then it continues, building what looks like a tree.
To answer your second question, almost every filed can have an application for
DT, at least for more "advanced" types, called Random Forest or Boosting, all you have to know in layman terms is that both try to find the best way to classify observation by averaging a lot of trees.
By this I mean that you have
trained lots of trees on the same data, and you take for each observation the major label (if most of them said that one has
lived the accident, then it probably is safe to say so).
This can shed some lights on a lot of applications as I was saying, from anomaly transaction detection in Banks, Medical Diagnose, some Regression problem, and even Handwritten Image Recognition.
To accompany @RLave's answer, another important part why decision trees are favored by the practitioners is their fast estimation (due to very simple algorithm, as described, since despite from cut-points no parameters need to be estimated; and the restrictions, if imposed, are computationally simple) despite relatively large data, plus fast implementation for execution. Nowadays the implementation part may not be that important, but note that it is easy to implement a fast decision tree model using several cheap detectors and a logic board.
So this adds a lot in practice, even though in most cases the trees are not the best accuracy wise. Random Forest and Boosting algorithms tries to tackle this, but either way they're often just a rough approximation of some complex nonlinear model.
NB: Would've commented this, but not enough rep.