Circuit parasitics, in a nutshell, are the resistive, capacitive and inductive elements of a circuit which arise as a result of practical components and PCB's exhibiting each of the above components and coupling between circuit elements.
Starting with resistors, all surface mount resistors have capacitance and inductance in addition to their set resistance where the capacitance is formed between the ends of the resistor and inductance through the resistive material. Note that these components can also be referred to as stray capacitance and inductance.
Similarly capacitors have resistive and inductive components and can be modelled as a series RLC circuit as shown below:
Finally, inductors have capacitive and resisitive components looking similar to a practical resistor.
Regarding coupling, the simplest to understand on a PCB is capacitive coupling with a simple example being the capacitance formed between all traces and the ground plane as there are two conductors separated by an insulative material, i.e. an effective capacitance.
Another common coupling mechanism is known as mutual inductance which is created when two current carrying loops are next to each other where the magnetic fields formed by each current loop interact with each other.
Note that all traces also have inductance known as self inductance, any current carrying wire or trace has inductance.
Another element which has many parasitic components is a transistor where there is capacitance between the base and emitter of a BJT as well as the collector and base and collector and emitter. All of these impact the frequency performance of the circuit and hence become particularly important to consider at high frequencies. Diodes also have capacitance between their terminals.
All of the above need to be considered for high speed designs or high precision circuits.