Officially, the sport is called weightlifting. However, in many countries weightlifting is an obscure sport and activity, so the term weightlifting is rarely adequate to distinguish the sport of Olympic-style weightlifting from powerlifting or bodybuilding, or even from weight lifting in the gym for fitness and strength. This is why the term Olympic Weightlifting is customary, as it the only barbell strength sport in the Olympics.
The sport of weightlifting consists of two lifts: the snatch and the clean & jerk. In the snatch, the weightlifter lifts the barbell from the floor to overhead in a single movement. In the clean & jerk, the weightlifter lifts the barbell first from the floor to the shoulders (the clean) and then from the shoulders to overhead (the jerk).
In a competition, each weightlifter has three attempts in the snatch and three attempts in the clean & jerk. The weight of the best of each lifts is added to create the weightlifter's total, and the best total wins.
The Olympic lifts, the snatch, clean and jerk, as well as variations such as the power snatch, power clean, power jerk, hang snatch, hang clean and more are often used by athletes to develop strength and explosiveness to improve performance in their sports.
The best way to learn the snatch and the clean & jerk is to work in-person with a qualified weightlifting coach that meets your specific needs, whether you want to be a world class weightlifter, or just learn the lifts for fun or to support other athletic activities.
If you don't have access to a good local weightlifting coach, or can't afford one, you can learn the Olympic lifts with this app, or this book.
You can also find hundreds of free videos, articles and an exercise library here to learn at no cost.
Finding a gym that allows Olympic weightlifting and has the appropriate equipment, such as weightlifting platforms, proper barbells and bumper plates, is much easier than it used to be. Many CrossFit or similar gyms offer weightlifting facilities and/or classes, and your national governing body for the sport (e.g. USA Weightlifting) should have a list of locations available.
You can find free and low-cost Olympic weightlifting training programs, free daily workouts, and online team training & coaching with a number of options here.
Olympic Weightlifting: A Complete Guide for Athletes & Coaches contains information on designing weightlifting training programs as well as many sample programs.
Generally you will need to be a member of the national governing body, e.g. USA Weightlifting, for the sport in your country to lift in sanctioned competitions.
Competitions can typically be found on the federation or regional organization websites, such as here.
You do not need to be a member of an official team/club, and do not need a coach; however, having a coach for a competition is extremely helpful. If you do not have one, you can find everything you need to know for competition here.
Olympic weightlifting competition internationally is governed by the International Weightlifting Federation (IWF), and in the US by USA Weightlifting. Your country will have its own federation that operates within the rules of the IWF.