An entity is the tabular representation of your domain class/object in the database and has an identity. In fact, an entity represents a single instance of your domain object saved into the database as a record. It has some attributes that we represent as columns in our tables.
For instance, in order to represent a Customer as an entity, it must have some attributes like CustomerId, CustomerName, Address and many more depending on the context. Consequently, these attributes form the columns of our Customer table. And therefore, each customer data that you save in the table, represents a customer entity.
Model
People often confuse entity with model. However, these two are quite different. A model typically represents a real world object that is related to the problem or domain space. While programming, we create classes to represent them. These classes, known as models, have some properties and methods (defining their behavior) in a particular domain space. For instance, in any customer oriented problem, we may have a customer class that has some properties and methods.
In some ORM (Object Relational Mapper) frameworks, a model is tightly bound to an entity. This means that every scalar property maps to an entity attribute. However, if you are familiar with Entity Framework, you must know that, not all properties in your domain class map to an entity column. And that's one way in which an entity is different from a model.
ViewModel
The term ViewModel originates from the MVVM design pattern. In the MVVM design pattern, it is the viewmodel that contains all the logic to handle the request/events generated by the view. We can say that a viewmodel in MVVM pattern is like a controller in MVC pattern.
However, there is one more side to it. A view has the responsibility of rendering data typically coming from an object. However, there are instances, when the data comes from two different objects. In such scenarios, we create a model class which consists of all properties required by the view. Different domain model instances then initialize this object. It's not a domain model but a viewmodel because, a specific view uses it. Also, it doesn't represent a real world object.
DataModel
The models in a particular domain space represent the real world objects. In order to solve a problem, these objects interact with each other. Some objects share a relationship among themselves and consequently, form a datamodel that represents the objects and the relationship between them.
In an application managing customer orders, for instance, if we have a customer and order object then these objects share a many to many relationship between them. The datamodel is eventually dependent on the way our objects interact with each other. In a database, we see the data model as a network of tables referring to some other tables.