Queen Bees will turn into several blocks of Beeswax if they've spawned in a fitting environment or are placed into a suitable environment. If the Biome or spot is unsuitable for Queen Bees to "grow" into Beeswax though, they will stay fallow and just stay like they are.
Currently Queen Bee blocks cannot be used for crafting.
How to obtain
Queen Bees can be randomly obtained as an occasional additional harvest when collecting Beeswax. Usually, one Queen Bee will be added after harvesting 1 to max. 30 blocks of Beeswax that has grown.
No Power Cells are required to pick up Queen Bees and/or harvest Beeswax, but since no Mining Cell loses durability when collecting such soft materials, it is recommended to use stronger Power Cells for faster progress.
Simply placing blocks of harvested Beeswax anywhere and picking them up again does not provide you with any Queen Bees. Instead you can definitely receive Queen Bees now and then from Beeswax that you've grown yourself by having placed Queen Bees on Wood or Leaves; and also from Beeswax that grows on trees that you have planted yourself.
When using Excavators on Beeswax, no Queen Bees will be obtained.
Growing Beeswax from Queen Bees
Queen Bees can be placed on blocks of Wood (Cragwood, Ashenwood, Elderwood, Wildwood, Weepwood, Autumnwood or Parchwood) or tree Leaves (Cragwood Leaves, Ashenwood Leaves, Elderwood Leaves, Wildwood Leaves, Weepwood Leaves, Parchwood Leaves, Cinnamon Autumnwood Leaves, Citrus Autumnwood Leaves or Mocha Autumnwood Leaves).
If you do this in a fitting environment, the Queen Bee block will transform into up to 6 blocks of Beeswax (into all "free" directions that are not occupied by any other block, liquid or the like) within 4 hours in real-time (or 3 hours and 45 minutes if you fertilize the Queen Bee), even if you're offline.
However, you cannot place Queen Bees on any types of blocks. If the block is unsuitable for Queen Bees, you will be notified with a message saying "Item cannot grow on this block type".
Since update R22 in September 2015 you cannot place blocks of Queen Bee just anywhere into the game-world any longer.
As mentioned above, Queen Bees can now only be placed on blocks of Wood and (since update R25) also on blocks of Leaves - except for Shorewood, Shorewood Leaves, Dark Wildwood Leaves, Snowy Elderwood Leaves, all types of Logs and all Corrupted Blocks - especially all types of Corrupted Wood and Corrupted Leaves.
Make sure that the Queen Bee is not fallow (not in an area that is too hot or cold, and not in specific unsuitable Biomes). The proper temperature ranges to grow Beeswax from Queen Bees is between 7 degrees and 109 degrees.
Fertilizing and time of growth
You can use fertilizer (Pigsy Droppings) on Queen Bees if you like, but it will only speed up the growth of Beeswax very little - just 15 minutes. You cannot fertilize Queen Bees if they are fallow.
Each Queen Bee (as long as it's not "fallow") will transform into a "hive" made of up to max. 6 blocks of Beeswax that will replace the Queen Bee within ca. 4 hours real-time.
If you have fertilized the Queen Bee with Pigsy Droppings, it will take ca. 15 minutes less (ca. 3 hours and 45 minutes).
You don't have to stay close or even on the game-world for Queen Bees to do their "work"; because Beeswax will also grow while you're offline.
When Queen Bees transform into Beeswax, the one block of Beeswax in the middle will replace the original Queen Bee and the rest will try to grow additionally on all 6 sides of the one block in the middle.
However if there are already any other blocks occupying any of these spots adjacent to the initially placed Queen Bee, then Beeswax cannot grow there. So if you surround a Queen Bee with Leaves, Wood or any other blocks all around, the Queen Bee can still transform into Beeswax in a suitable area, but only the size of 1 block, if the surrounding space is already occupied and will block Beeswax from growing there.
Fitting spots for Queen Bees to grow
Queen Bees do not need any light to grow and such can also grow well underground, even on the Corruption layer if planted on fitting blocks of Wood or Leaves (not on corrupted ones). Only the Lava layer will be just too hot of course to let Beeswax grow, but it might also be unfitting as a Biome itself, because no matter how much you cool the area down, Queen Bees will stay fallow (just like Seeds of Crops).
As mentioned, 6 degrees are too cold and 110 degrees or more are too hot for Queen Bees to grow into Beeswax. In Jungles and Savannahs, you can try to search for cooler spots, like in Caves or somewhere higher up, even on platforms built in the sky (but not too high, since the air can become too cold to grow anything at an altitude of 150, sometimes even around 140).
Swamplands and their neighbouring areas can be unsuitable for Queen Bees because of the Bog Water pools or because of the Biome per se. However, it might still be possible to grow Queen Bees in Caves just below Swamplands (Fossil layer, Stalactite layer, even Corruption layer) - but not on platforms hovering high above.
Differences between "self-grown" Beeswax and "generated" Beeswax
Lots of patches of Beeswax are already part of each newly created game-world, and more will grow over time additionally here and there on nearly every kind of tree except for palm-like Shorewood.
Initially existent Beeswax (that comes with world "generation") can be found in patches of 7 blocks and sometimes two or even more of these patches will be clogged together - not only on top of trees, but also embedded right in the middle of leaf canopies.
Patches of Beeswax that you can grow can only have up to 6 blocks, because one possible spot for the maximum of Beeswax blocks to grow is already taken by the block of Wood or Leaves that the Queen Bee has been placed on.
Additionally to that, Beeswax that is part of the template game-worlds can often be found in areas that are unsuitable for Queen Bees to grow into Beeswax. So it's very common that you cannot simply place a Queen Bee on the same spot on tree tops where you've just harvested Beeswax from.
After world "creation" more Beeswax grows automatically over time from Queen Bees that spawn randomly on tree Leaves on the game-world. These "regrowing" Queen Bees can sometimes appear in areas that are too hot or cold or otherwise unsuitable for Beeswax to grow, so the Queen Bees will just stay fallow there.
If such regrowing Queen Bees grow into Beeswax, these patches also cannot consist of more than 6 blocks of Beeswax, but often even less if there's not enough free space for all of the possible blocks of Beeswax to grow.
The "Pro" game-world option "more regrowth" will make more Queen Bees spawn all over the world (the most in areas close to where player characters spend their time), but will not make them turn into Beeswax any faster.
When Queen Bees stay fallow
Queen Bees that have spawned or are placed in too hot or too cold biomes, on very high altitudes or on/near the ground of Swamplands will stay fallow no matter what. You cannot fertilize fallow Queen Bees (the same goes for fallow Seeds and fallow tree Saplings). It's hard to tell where Queen Bees will grow or not. They are more "fickle" than Seeds of Crops and tree Saplings. Since initially created patches of Beeswax have been generated on template worlds under different circumstances and "rules", newly placed Queen Bees might often not grow on spots or in whole Biomes where you can harvest a lot of Beeswax.
Just take care not to plant Queen Bees (on Wood or Leaves) in too cold or too hot environments, neither on altitudes too high up nor on the ground of Swamplands.
In such environments Queen Bees will stay fallow (you will see this word written in red letters when taking the Queen Bee into your focus) and of course fallow Queen Bees will not produce Beeswax under any circumstances, but will simply stay Queen Bees.
It seems like it does not matter how much Beeswax has already grown in the area, on surrounding trees or the like. So you can plant a lot of Queen Bees next to each other and they will all grow into Beeswax if the area is suitable and won't make them stay fallow.
Display (Wood Planters, Flower Pots)
Please note that Queen Bees that you put on display in Wood Planters, Flower Pots, on Wall Shelves, Placemats and/or Hidden Temple Altars will not grow into Beeswax there, but simply be preserved. The same goes for Seeds of Crops and tree Saplings too.
When placing Queen Bees close to (or especially directly above) torches or any other items with open flames or fiery blocks like Hardened Lava, then Queen Bees can easily start to burn in hot environments that make a heat meter show up, like Oceans, Jungles or the Lava layer, but also in much cooler Swamplands.
Placing several torches or heat sources with open flames next to each other can even raise the chances to set flammable blocks on fire. Liquid Lava or Fire Bombs will immediately set Queen Bees on fire.
If Queen Bees start burning, the flames are then able to spread to other nearby inflammable blocks, even across gaps of 1-2 blocks on worlds and player claims where the "fire spreading" option is enabled. This way a whole wooden building or forest (especially Wildwood, Parchwood and Shorewood) can burn down in the worst case.
Fire cannot be extinguished by using liquids, but it can be stopped from spreading by claiming the area and making sure that the claim (advanced) option "fire sim/spreading enabled" is disabled (by default). Fire Bombs will have no effect on these claims either.
It is also possible for owners of the game-world (F2P players included) to toggle the spreading of fire on a whole game-world in the basic world options ("edit world" and "disable fire spread"). Another option to stop fire from spreading would be to create forest aisles/swaths wide enough so that the flames cannot leap over.