Power systems have typically been designed to be reliable to expected, low-impact high-frequency outages. In contrast, extreme events, driven for instance by extreme weather and natural disasters, happen with low-probability, but can have a high impact. The need for power systems, possibly the most critical infrastructures in the world, to become resilient to such events is becoming compelling. However, there is still little clarity as to this relatively new concept. On these premises, this paper provides an introduction to the fundamental concepts of power systems resilience and to the use of hardening and smart operational strategies to improve it. More specifically, first the resilience trapezoid is introduced as visual tool to reflect the behavior of a power system during a catastrophic event. Building on this, the key resilience features that a power system should boast are then defined, along with a discussion on different possible hardening and smart, operational resilience enhancement strategies. Further, the so-called ΦΛEΠ resilience assessment framework is presented, which includes a set of resilience metrics capable of modeling and quantifying the resilience performance of a power system subject to catastrophic events. A case study application with a 29-bus test version of the Great Britain transmission network is carried out to investigate the impacts of extreme windstorms. The effects of different hardening and smart resilience enhancement strategies are also explored, thus demonstrating the practicality of the different concepts presented.