It is said that 8051 has an on-chip oscillator but requires external clock to run it. What may be the internal circuit for on-chip oscillator, how it runs after connecting crystal oscillator? Also what is the purpose of external capacitor with crystal oscillator?
"The original 8051 architecture was designed such that crystal, or RC oscillator, or an external clock generation circuit can be used as clock input to 8051"
Yes. Exactly.
So it is incorrect to say that it requires an external clock to run it.
Sure, an external clock can be used, but it is not required.
That was my point.
"RC oscillators ... are AC circuits"
As is a crystal oscillator - the crystal provides the resonant element, as does the RC in an RC oscillator.