I am looking for a good scripting language to link to my program.
I am looking for 2 important attributes:

Scripting language should be hard linked into the executable (not requiring 3rd party 
installations). This is important to me to simplify distribution.
Scripting should allow some run-time debugging option (When running a script inside my program I would like to easily run it inside a debugger while it is running in the context of my program)

Can python,lua or some other language supply me with this?
Both Lua and Python can provide the features you mention, so choosing one of them will depend on other criteria.
Lua is a lighter weight solution, it will have a much smaller disk footprint and likely a smaller memory overhead than Python too. For some uses it may be faster. Python has a much richer standard library, more mature third party libraries and a more expressive language.
Both have been embedded into major applications. Python can be found in Blender, OpenOffice and Civilization 4. Lua can be found in World of Warcraft and Adobe Lightroom. I'd recommend looking at a few tutorials for each and the facilities available to embed them in your application and just choose the one that fits your brain best.
Scripting Languages