My (non-standard) Layman definitions

• POCO – Plain Old %Insert_Your_Language% Object. A type with no logic in it. It just stores data in memory. You’d usually see just auto properties in it, sometimes fields and constructors.
• Domain object an instance of a class that is related to your domain. I would probably exclude any satellite or utility objects from domain object, e.g. in most cases, domain objects do not include things like logging, formatting, serialisation, encryption etc – unless you are specifically building a product to log, serialise, format or encrypt respectively.
• Model object I think is the same as Domain object. Folks tend to use this interchangeably (I can be wrong)
• Entity a class that has id
• Repository a class that speaks to a data storage from one side (e.g. a database, a data service or ORM) and to the service, UI, business layer or any other requesting body. It usually hides away all the data-related stuff (like replication, connection pooling, key constraints, transactions etc) and makes it simple to just work with data
• Service software that provides some functionality usually via public API. Depending on the layer, it can be for example a RESTful self-contained container, or class that allows you to find a particular instance of needed type.

Original answer

These are terms that are largely used in (Distributed) Domain Driven Design. They are not the same. The term model Object can be used as a synonym to the domain object.

Domain Objects. Objects from the business specific area that represent something meaningful to the domain expert. Domain objects are mostly represented by entities and value objects. Generaly speaking, most objects that live in domain layer contribute to the model and are domain objects.

Entity. An object fundamentally defined not by its attributes, but by a thread of continuity and identity. (Meaning it must have Id)

POCO. A simple object without complicated logic, usually it has just a few properties and is used with ORM or as a Data Transfer Object

class Person – Entity and POCO, instance of this class is Domain Object
class PersonService – Service
class PersonRepository – Repository