Business Add-Ins are a new SAP enhancement technique based on ABAP Objects. They can be inserted into the SAP System to accommodate user requirements too specific to be included in the standard delivery. Since specific industries often require special functions, SAP allows you to predefine these points in your software. 

As with customer exits two different views are available:

In the definition view, an application programmer predefines exit points in a source that allow specific industry sectors, partners, and customers to attach additional software to standard SAP source code without having to modify the original object. 

In the implementation view, the users of Business Add-Ins can customize the logic they need or use a standard logic if one is available.

In contrast to customer exits, Business Add-Ins no longer assume a two-level infrastructure (SAP and customer solutions), but instead allow for a multi-level system landscape (SAP, partner, and customer solutions, as well as country versions, industry solutions, and the like). Definitions and implementations of Business Add-Ins can be created at each level within such a system infrastructure.

SAP guarantees the upward compatibility of all Business Add-In interfaces. Release upgrades do not affect enhancement calls from within the standard software nor do they affect the validity of call interfaces. You do not have to register Business Add-Ins in SSCR.

The Business Add-In enhancement technique differentiates between enhancements that can only be implemented once and enhancements that can be used actively by any number of customers at the same time. In addition, Business Add-Ins can be defined according to filter values. This allows you to control add-in implementation and make it dependent on specific criteria (on a specific Country value, for example).

All ABAP sources, screens, GUIs, and table interfaces created using this enhancement technique are defined in a manner that allows customers to include their own enhancements in the standard. A single Business Add-In contains all of the interfaces necessary to implement a specific task.

The actual program code is enhanced using ABAP Objects. In order to better understand the programming techniques behind the Business Add-In enhancement concept, SAP recommends reading the section on ABAP Objects.

What is difference between badi and user-exists?
What is difference between enhancements and user-exists? and what is the full form of BADI?

I have another doubt in BDC IN BDC WE HAVE MSEGCALL (i did not remember the > correct name) where the error logs are stored, MSEGCALL is a table or structure.

What is the system landscape?

1) Difference between BADI and USER-EXIT.
    i) BADI's can be used any number of times, where as USER-EXITS can be used only one time.
       Ex:- if your assigning a USER-EXIT to a project in (CMOD), then you can not assign the same to other project.
    ii) BADI's are oops based.

2) About 'BDCMSGCOLL' it is a structure.  Used for finding error records.
3) Full form of BADI 'Business addins'.
3) System land scape will be depends on your project 
    Ex:- 'Development server'-->'Quality server'---> 'Production server'...... 

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