Why An LC oscillator needs a fraction of the tank's output fed back to the input in phase to sustain oscillation, and the different classic topologies are named for how that tap is made. In a Colpitts oscillator the tank uses two series capacitors across the coil, and feedback is taken from the junction between them, so the capacitive divider sets the feedback ratio.
Watch out The tapped coil describes the Hartley oscillator, which splits the inductor instead of the capacitance; link coupling through a separate winding is the Armstrong arrangement. A neutralizing capacitor does the opposite job, cancelling unwanted feedback in an amplifier.
Colpitts has a C, so Capacitors; Hartley has no C, so the coil is tapped; Armstrong uses a transformer link.