Ever since the ADC support lane came into existence, there was an understanding about how supporting worked, both from a design and from a player standpoint.
The idea was that there were three main roles supports played in-lane. Sustain, All-In, and Poke.
Sustain supports were the classic supports who had the ability to heal others based on the iconic white mage style supporting from various RPGs. Examples include Janna, Sona, and Nami. However, it is of note that this playstyle, once established, became extremely unpopular with Riot, who refused to release another champion that could heal others beyond 2011 until Nami, and even then they tied her damage to her heal and gave it an insane mana cost. Bard is the only other champion from 2012 on to have a heal in his kit.
Poke supports were the other type of "classic" supports who had the ability to deal spikes of magic damage from a distance in order to push the lane and punish out of position carries, disrupt farm, and prevent ganks with the cc often tied to their poke. Examples include Lulu, Nunu, and Kayle.
All-in supports originally weren't that well supported, but have since become popular with the creation of Thresh, Leona, and Braum. They are usually melee supports that play aggressively and chain cc in order to take out squishy targets and help their junglers successfully gank. Unlike sustain supports, who are generally masters of disengage, and poke supports, who are masters of poking and sieging, all-in supports were designed to be powerful engagers, a role that once was the duty of the jungler or the top laner.
So, this was the standard. Sustain countered Poke, Poke countered All-In, and All-In countered Sustain. It was a system that worked pretty well, and then Riot started messing with it.
As part of their ongoing project (The Team Builder system) Riot made some changes to support itemization in Season 3. They introduced items that gave supports much more gold than before, as well as scalings to help fit this gold. This did two things for the supports in the game:
1) It gave AP scalings to poke and sustain supports, as well as reducing their bases. This resulted in poke and sustain supports heading mid lane in order to farm as much as possible and use their new scalings to oppress the enemy, as their peel designed for carries worked just as well for themselves, and their lower base utility and damage meant they had a weaker early game in a duo lane to help.
2) It gave All-In supports both more gold to buy tank items (already cheaper than offensive items for balance reasons) as well as new-found utility for building this survivability.
Because these sustain and poke supports went mid lane, Riot had to go in and nerf the champions themselves. Lulu and Zyra both received hard nerfs to their utility (including a questionable nerf to Lulu's shield and Pix duration which hurt support Lulu more than mid Lulu, who leveled Glitterlance first). Soraka and Sona were both reworked, having their base power lowered immensely and losing a lot of their old identity while also reducing their survivability (and thus their ability to function in a solo lane) substantially.
The imbalance between the buff to All-In supports and the nerf to Poke supports led to a hole in bot lane that ended up being filled by mages. Annie and Morgana became the new poke supports, hard counters to All-In supports whom Riot could not easily nerf without affecting their power in the mid lane. Eventually, Riot did nerf Annie (mostly) out of bot lane while buffing her mid lane power, and gave Morgana some love taps that did not really affect her play rate in any sizable way, because she was still so necessary to deal with Thresh, Blitzcrank, and Leona.
To Riot's credit, Thresh, Blitzcrank, Leona and Braum have all received nerfs in order to make them easier to deal with, with Riot specifically hitting Leona's base health regen and stating plainly that the proper counter to her was poking her out of lane. However, with poke supports practically removed from bot lane this has become a non-issue. Leona is easier to kill in team fights now, sure, but in lane she remains a problem. A problem that the jungler now has to deal with because supports lack the power to do so themselves.
Then came the ADC itemization overhaul. The nerfing of Bloodthirster and the power boost of Infinity Edge, as well as the changes to how mana regeneration works (which nerfed Tear of the Goddess, pretty much the only item ADCs could use to keep their mana pools somewhat afloat) kicked ratio ADCs out of lane and made auto-attack ADCs the new gods.
Ratio ADCs like Lucian, Varus, Ezreal and Caitlyn dealt with all-in supports quite well, due to their longer range, poke and self-peel. However, the nerfs to itemizations saw all their win rates drop and caused them to leave bot lane in favor of stronger auto attack champions like Jinx, Tristana and Vayne (and then Tristana got reworked). Admittedly, since then Graves has seen a resurgence as the premier ratio ADC, but this did not happen quickly enough to prevent the change that this itemization shift brought about in supports.
Poke supports depend on ratio carries to help them with their game, which is "stop the enemy carry from CSing and win a war of attrition". Meanwhile, All-In supports **love** Auto Attack carries, because their extended CC chains allow for long, one-sided trades, which Auto Attack carries deal much more sustained damage with. Thus, the itemization changes for ADCs in 4.10 were the nail in the coffin for poke supports, and we've seen nothing but All-In supports (and Nami and Morgana) ever since.
**TLDR** Sustain counters Poke, Poke counters All-In, All-In counters Sustain, but a lot of changes Riot has made has left Poke and Sustain supports unplayable, leading to the current state of bot lane where All-In supports reign supreme.