I have a map, with point layers A, B and C.
A is the main point layer. I would like to make something like this but I have no idea how I am going to do this.
When I start an edit session, click on a point in layer A, delete it, all surrounding object within a radius of r meters in layer B and C would be deleted as well.
What you are asking could easily be accomplished interactively (manually) by first selecting the desired point feature in layer A, doing a Select by Location to select the features of layers B and C that intersect the selected feature in layer A using a buffer ("search distance" on the Select by Location dialog) of r meters, and then deleting all of the selected features.
A similar approach could be also be used with geoprocessing tools in ModelBuilder or a Python/arcpy script tool to automate the process, though you would need to add a check that ensures that layer A has exactly 1 feature selected before continuing or you might end up deleting everything.
Also, in ArcGIS 10, if you set up a script tool or model with no parameters and place it on a toolbar, it will run immediately without user interaction.
If you are using ArcMap 10.1 or later, then you could use a Python AddIn to do this. To learn about Python AddIns see What is a Python add-in?:
An add-in is a customization, such as a collection of tools on a toolbar, that plugs into an ArcGIS Desktop application (that is, ArcMap, ArcCatalog, ArcGlobe, and ArcScene) to provide supplemental functionality for accomplishing custom tasks.
ArcGIS 10 introduced a new and innovative Desktop add-in model, making it easier for you to customize and extend the ArcGIS Desktop applications. The new add-in model provides you with a declarative-based framework for creating a collection of customizations conveniently packaged within a single, compressed file with a .esriaddin file extension.
