When and why is it very effective to use Zhonya's Hourglass?

Most ap champs can use Zhonya.

The ones that make the most use of it are the ones like Swain that have an ability that keeps going even when using it.

Swain, Morgana, Kennen, Fiddlestick (ults) and Karthus (defile) are the ones, of the top of my head, that have such abilities that will keep going while under Zhonya protection.

It can also be used on squeesh ap champs that use it for the protection only and that they need to be in the middle of the fight to deal the most damage, like Fizz and Katarina. It's a life saver. It may be used on Kassadin but with his escape ability, it usually not as useful. It will be seldomly used on Annie, Brand and the other ap champs that usually are not in the middle of the fight in case they get there and needs time for another cycle of spells.

Other ap champs can still use it but some don't generally use it, like Diana and Orianna. You won't see many people running Zhonya on those 2.

Any champion that has an ongoing high damage ability that deals damage over time. For example, Fiddlesticks ultimate ability allows him to teleport into a group of enemies with an area of effect damage over time attack. The weakness of this ability is that Fiddlesticks is a low-HP squishy champion. The couple seconds of invulnerability from Zhonya's allows him to deal a proportionately high amount of extra damage, often leading to multikills.