First of all, what is a "model"? It sounds quite complicated, but actually everyone uses models to understand the world and ourselves: a model is a simplified way of understanding the reality. For example, to know a city, we draw a map--that is a model of the city. To visualise a plane, we build a toy plane--that is a model of the real plane. And to understand how the economy works, we formulate a mathematical equation system to mimic people's economic behaviour.
But why do use a "model"? Take the toy plane and the map of a city as examples. If we replicate every single detail of the plane/city, then we end up with a 1:1 toy/map--which are just useless! Yes, the ultimate reason for using a simplified version (the model) of the complicated reality is because we only care about something, not everything, of the reality. Too much trivial details will prevent us from seeing the information to guide our decision making. Obviously, the toy/map are NOT equal to the reality, but it has everything we need them for: to visualise and to navigate. Similarly, an economic model is not to depict every trivial detail of the economy, but to understand the economic phenomenon (positive analysis, like "visualisation") and to prescribe the economic policy (normative analysis, like "navigation"). "All models are wrong, but some are useful." (Box, 1976)
As a result, all models should have a particular purpose, and for different purposes models should be different. It is like we need tourist map for tourism and resources map for mining. Now that we explained what is a model and why we need a model, let's look at some technicalities of how to build a model.
There are two types of models in the methodological sense. One is theoretical models (or economic models), including those you learned in microeconomic theory (e.g. consumer theory, producer theory, equilibrium theory, etc.) and macroeconomic theory (e.g. New Classical, New Keynesian, etc.), which are all based on a fundamental doctrine since Adam Smith: individuals are self-interested. The implication of this fundamental assumption about human being is that individuals (consumer or firm or government) are "optimisers". (In contrast, Behaviour Economists do not advocate this assumption of "optimisers", but explaining economic behaviour using psychological theory.) Usually, there are some endogenous variables (y) under the optimisers' control, some exogenous variables (x) beyond the optimisers' control, and they are linked by some "deep" parameters (theta). Note that we call theta "deep" parameters following Lucas critique (1976) because these parameters governing the economic behaviour do NOT change when policy or exogenous variables change. Which variables should be included as endogenous is determined by the Research Question. For example, if you are studying business cycles, then monetary policy is obviously a key variable. It would be less important if you are studying long-run economic growth. To connect x, y and theta, a typical economic model is made up of some first order conditions derived from some optimisation problems, resulting in something like F(x, y, theta) = 0. Since the RBC revolution in the 1980s, the mainstream macroeconomic models are also "microfounded" in line with microeconomic models. These first order conditions F(x, y, theta) = 0 are termed as "structural form" of the model, because it is an unsolved system of equations. By solving it, the final form or the "reduced form" of the model should look like this: y = f(x, theta). In other words, each endogenous variable is expressed in terms of only exogenous variables and deep parameters. If you are lucky, you will be able to obtain the reduced form analytically, but most of the time the solution is obtained numerically.
The other type is empirical models (statistical or econometric models), such as linear regression model, probit, tobit, Poisson, Cox; ARIMA, ARCH, VAR, VECM; fixed effects, random effects, ... Usually, it takes a form like this: y = h(b*x + e), where y is the endogenous variable vector, x is the exogenous variable vector, b is the parameter vector, e is the error term and h(.) is the link function (linear or nonlinear, univariate or multivariate). These econometric models are closely related to economic models, because usually the "x" vectors in both models are identical. However, the "deep" parameters (theta) and the regression coefficients (b) are not identical. In some simple cases where f(.) and h(.) are both linear, b is a function of theta, and if you are lucky you will be able to identify or recover theta from the estimated b. However, in general, if f(.) and h(.) are different due to the model specification, it is very difficult to map theta with b.
The econometric models are very powerful and succinct ways of describing and summarising the data features, but they are NOT equal to the economic models in two senses. Firstly, it over-simplifies the relationship between y and x using a linear or log-linear function form. Secondly, the regression coefficients b are not well defined to correspond to the economic models (theta). It is dangerous to think that econometric models can be used to verify or falsify the economic models, because they can be very different things. It is ironic that in macroeconomic literature, most models have microfoundation nowadays, i.e. they are economic models both theoretically and empirically. However, in microeconomic literature, most empirical models do NOT have strict microfoundation, i.e. they are econometric models. I am currently working on filling this gap between microeconomic and microeconometric models, i.e. providing microfoundation for microeconomics!
These models need to be confronted to data by empirical techniques.