Bootstrapping comes from the stories about Baron von Münchausen who got stuck in quicksand and lifted himself by grabbing his bootstraps; that the output transistor "lifts its own control voltage" is what is meant with "bootstrap" functionality.
A) In analog circuit designs a bootstrap circuit is an arrangement of components used to boost the input impedance of a circuit by using a small amount of positive feedback, usually over two stages. This was often necessary in the early days of bipolar transistors, which inherently have quite a low input impedance. The need for such arrangements has largely been alleviated by the use of modern field effect transistor designs, except when ultra-high input impedances are required. Note that because the feedback is positive, such circuits usually suffer from poor stability and noise performance compared to ones that don't bootstrap.
B) A N-MOSFET/IGBT needs a significantly positive charge (VGS > Vth) applied to the gate in order to turn on. Using only N-channel MOSFET/IGBT devices is a common cost reduction method due largely to die size reduction (there are other benefits as well). However, using nMOS devices means that a voltage higher than the power rail supply (V+) is needed in order to saturate the transistor and thus avoid significant heat loss.
A bootstrap capacitor is connected from the supply rail (V+) to the output voltage. If the capacitor is polarized then the orientation of the capacitor is as follows: Anode(marked with ‘+’)→(V+) and Cathode (marked with ‘-’)→Output. In other words, the capacitor should be between the output (source of an N-MOSFET) and (V+). Usually the source terminal of the N-MOSFET is connected to the cathode of a recirculation diode allowing for efficient management of stored energy in the typically inductive load (See Flyback diode). Due to the charge storage characteristics of a capacitor, the bootstrap voltage will rise above (V+) providing the needed gate drive voltage.
A MOSFET/IGBT is a voltage controlled device which, in theory, will not have any gate current. This makes it possible to utilize the charge inside the capacitor for control purposes. However, eventually the capacitor will lose its charge (due to parasitic gate current), so this scheme is only used where there is a steady pulse present. This is because the pulsing action allows for the capacitor to discharge (at least partially if not completely). Most control schemes that use a bootstrap capacitor force the high side driver (N-MOSFET) off for a minimum time to allow for the capacitor to refill. This means that the duty cycle will always need to be less than 100% to accommodate for the parasitic discharge unless the leakage is accommodated for in another manner.
C) Also in SMPS (Switch Mode Power Supplies) a leakage resistance can be used to trickle charge supply rail for the control circuit to start it oscillating. From then on it sources power from its own output.