ArcObjects is a set of COM-based APIs from ESRI Inc. that are used in several of their products including ArcGIS Desktop, ArcGIS Engine and ArcGIS Server.
ArcObjects is a set of platform independent, component based geographic data models written in C++, that allow developers to extend the ArcGIS family of applications.
In the ArcGIS 10.X suite APIs can be used on several platforms including .NET (VB.NET and C#), Java and C++.
In the ArcGIS 9.X suite of products VB6 was previously supported, but its use has been deprecated in favor of VB.NET/C#.
For more details see What are ArcObjects?