The Loved beatmap category is currently managed by Project Loved. However, this wasn't always the case. Project Loved is the third main system that has promoted beatmaps to Loved; prior to it, there were other methods to do so, as well as long periods of stagnation for the Loved category.
On 2015-02-02, the feature request New beatmap category for good unrankable mapsets was posted. It detailed another way outside of the ranking process for maps to have leaderboards, and quickly gained traction as players showed concern about the ranking process becoming too strict and not allowing for some maps to house competition.
Another reason this request became popular is that very unconventional or game-breaking maps could have never fit the ranking criteria. Players were interested in competition on some of these maps, such as ones featuring "2B" styles, Aspire-like gameplay, or TAG co-op elements.
By late 2016, this feature request held the second-most stars in the forum's history with nearly 13,000 priority.
When Loved was first created, it aimed to address everything in the feature request. Polls were opened by Ephemeral on the forums where anyone could help decide which maps would become Loved. Details of the voting system were explained in the Show Your Love! news post.
The opening rounds of voting saw great success as the community enjoyed participating in this new feature. Unfortunately, it wasn't long before some important and difficult concerns came to light. By the fourth round of voting, a common complaint was that while the community could vote in the polls, it didn't feel like anyone had much influence in deciding which maps got to the polls in the first place. Many mentioned that the infrequency of the voting wasn't supportive of the idea behind Loved. It was also cause for concern that maps were effectively competing against each other in this voting, despite all of the maps being popularly praised in some way, and thus all apparently suitable for Loved.
Following feedback received during the fourth round and in a general poll about the state of Loved opened around a month before it, Ephemeral (and other members of the osu! team) decided that the votes in this format would not be sustainable in the long term. He strongly supported the idea of using favourites as a metric to decide which maps would be admitted to Loved.
Just a few weeks after the fourth round of community voting, Loctav posted an announcement describing Loved's next system of beatmap admission. Instead of a select set of maps being put in polls, every map became eligible for Loved, so long as it had at least 30 favourites and 100 star priority (SP).
The implication of the high SP requirement was that the mapping and modding community would be involved in the process of moving maps to Loved: SP could only be added to maps by spending kudosu!, a currency obtained by modding other maps. This immediately became a controversial point with the wider community, who thought that Loved was supposed to be about getting leaderboards on popular maps, not just the ones that "mappers" liked.
Over the course of this system's stay, some of the newly-Loved maps became points of controversy, with players citing that it was exactly what they had feared about using a great amount of SP and a comparatively low amount of favourites to get a map Loved. On the other hand, the minority that liked this system also had complaints that the SP requirement was absurdly high; getting 100 SP required such a huge time and effort investment that it wasn't worth contributing to the Loved cycle. These opinions and more can be found in the thread of Ephemeral's open letter to the community that he hoped would provide the insight needed to solve the apparent issues with Loved.
Ephemeral posted some updates later on in his forum thread, but no action was taken and Loved became inactive during June 2017.
The second episode of osu!'s Coffee Hour talk show featured HappyStick, Ephemeral, and pishifat discussing the past and future of Loved. Near the end, Ephemeral suggested that the original community voting could be brought back in some form, since the community seemed to like it better than the star priority and favourites system, provided the selection of maps was good. The question of how to choose those maps was left unanswered in this show.
By the end of September, Toy had collaborated with Ephemeral to create a new version of the community voting called Loved Captain's Pick. At first, it was nearly identical to the old voting, with the exception that the maps in the polls were chosen by "captains", trusted community members that had a good sense of what the rest of the community wanted to see.
After a few rounds of this, Toy decided to use a slightly different voting format where the maps would no longer have to compete against each other. Every map had its own poll from this point on, asking the community whether or not they wanted that specific map to be Loved. If a threshold percentage of voters said "yes", it would pass the voting. Due to requiring so many forum polls, a new forum was made specifically for Loved voting, and it was from this point forward that it became more formally known as "Project Loved".
Project Loved is the current system being used to move maps to Loved. More details and information about how to get beatmaps up for voting in Project Loved can be found in its wiki article.