There's an old aeronautical proverb: "Take off is an option, landing is mandatory". Anyway, pilots of the powered aircraft can choose the place and time of landing, with their options usually limited by available fuel, aircraft performance and aircraft technical status. Even after they have made their decision about the landing spot, they can change their minds; throttle forward, go-around and climb to the safe altitude for another attempt. Principle is the same for the piston singles, fighter jets or airliners; whenever the pilot estimates it is safer to give-up landing attempt than to continue to landing, go-around is executed. General outlines are also the same; first the adequate power or thrust for climb is assured, high lift devices (flaps and slats) are retracted to lower setting, undercarriage (if retractable one is fitted) is retracted when aeroplane achieves positive climb and further flightpath depends on future plans. Fighter jet will probably go into traffic pattern, Cessna too but its circuit would be lower, narrower and flown at quite lower speeds. Airliners usually follow the published missed approach procedure.
What the go-around looks like in the cockpit of the typical airliner?
When the pilot in command decides that go-around should be made, or confirms the air traffic control instruction, pilot flying (PF) adds some power, activates the flight director go-around mode (by setting the trust levers fully forward on Airbus or pressing TOGA button on everything else) and requests retraction of the flaps to lower setting. Pilot monitoring (PM) confirms that go-around trust is set (or sets it herself) and selects the flap partial retraction. When the aeroplane is established in climb, PM calls out "Positive climb" (or "Positive rate) and PF requests gear retraction. Aeroplane now follows the missed approach procedure, published for that specific runway and the type of approach, PM informs the air traffic control about go-around. Trust is reduced to climb setting at the acceleration altitude and flaps are (gradually) retracted. Now comes decision time; whether to wait in holding pattern, try another approach or divert to an alternate aerodrome. Some airliners can perform missed approach fully on autopilot, some can use AP only in certain circumstances while some (smaller and cheaper) have to be manually flown during go-around. Nowadays it is widespread policy that even when the captain is pilot flying, if the aeroplane has descended below 1000 ft height above the aerodrome, if the co-pilot concludes that safe landing is no longer possible, he will waste no time explaining what's going on but rather he will just call out: "Go-around!". After this, captain must have very, very, very good reason not to perform one.
What's the lowest height to initiate go-around?
Almost always – zero. Even when all the wheels are rolling along the runway and ground spoilers deploy, there is still an option of pushing the trust levers forward and taking off again. The decision point, after which the aeroplane is committed to stop, is trust reversers deployment.
What if one engine fails during go-around?
No big deal, provided the aeroplane is twin or more engined. Before every landing (well, at least in the airlines) performance calculation is made to check whether the published missed approach procedure can be flown with engine failure at the most critical moment or the special procedure has to be flown. Engine failure just as the go-around initiated and missed approach after one-engine-out approach are regularly practised in the simulator, at least every six months.
What are the typical reasons for go-around?
The most common one used to be failure to achieve visual contact with the runway environment at the minimum descent altitude, the altitude below which instrument guidance is no longer safe and landing must be continued by visual reference. Advancement of aerodrome meteorological services and radio navigation aids has significantly reduced such occurrences. Until CoViD pandemic, heavy traffic meant short approach spacing and if the preceding traffic didn't vacate the runway rapidly, following traffic would need to go around. Due to frequent incidents (or worse) when pilots would try to save the landing by low height manoeuvrings, stable approach policy was implemented. It means that at 1000 ft above runway threshold, aeroplane should be configured for landing, at approach speed, with typical approach thrust and landing checklist complete. If it cannot be achieved or if the airplane gets significantly destabilized (usually due to turbulence) below 1000 ft, it's better to go-around and return for another try. Touchdown has to be achieved within the first third of the landing runway or first 900m, whatever is shorter. If it's not – go around. ATC has to issue the go-around instruction any time the controller estimates it is not safe to land. Last time I was ordered to go-around it was due to two calf-sized dogs that managed to dig under the airport fence and were currently chasing each other merrily on the runway.
How is the decision made whether to try another approach or to divert to alternate?
It really depends on the reason for go-around and general airport situation. If the go-around was due to weather and it is worsening, it is no brainer: fly to the place where it is nicer. If the go-around was due to someone being on the runway at the time he wasn't supposed to be there; as soon as he clears the runway, we'd try again.
Is the go-around risky manoeuvre?
As every other; no, if performed properly, which is the most common case. It's often practised in simulator and while in the pre-pandemic aeronautical world they were performed aplenty on a daily basis, individual pilot might perform it for real once a year or even less frequently than that, so occasionally someone messes up. Thanks to the internet and its real-time flight tracking apps, every unusual manoeuvre gets its three minutes of fame on social networks and more spectacular ones get picked up by the news (or "news") portals.
I guess that covers the basics of go-around. Feel free to ask about it in the FB comments.