Lime is a new Material in Farming Simulator 19. It can be spread on fields to remove the "Needs Lime" state, increasing the crop yield by 30% of its normal value. This is the only use for Lime in the game.
Fields gain the "Needs Lime" state after three harvests have been completed (of any crop). This is seen as a cyan color on the field in the "Soil Composition" map mode, and is also reported on the field information box when looking at a field up-close. In the base game, Lime can only be spread by two models of Fertilizer Spreaders (the third cannot spread Lime).
Lime may be purchased in loose form at Lime Stations on the map for a relatively cheap price. Alternatively, Lime can also be purchased in Pallet form at the Store for a higher price. Lime can be carried in bulk inside certain Tipper models, or dumped as Heaps on the ground. There is no way to store Lime in Placeables or Auger Wagons, making long-term storage and distribution somewhat awkward - though Fertilizer Spreaders can unload their Lime contents in Pallet form. The material itself is rather heavy, making transportation and loading awkward as well.
NOTE: You can turn off the "Needs Lime" state altogether through the options menu.
"Needs Lime" State
So long as the "Lime" option is turned on in the options menu, the game keeps track of how much Lime is left on each piece of field on the map. Spreading lime on a piece of field resets the value to 1, regardless of the state of that field.
Each time a Crop is harvested on that piece of field, the value is reduced by 1/3. Once it reaches 0 (after three harvests), that piece of field now requires Lime.
So long as the Lime value of a piece of field is greater than 0, that piece of field will yield 30% more crop than the base value. Once the field "Needs Lime", that bonus is lost.
The Lime value is tracked separately for each piece of field. It is quite possible for part of a field to require Lime while the rest does not require it.
Purchasing Lime
Lime may be purchased in two different ways, each providing the material in different forms and for a different price.
Lime Stations
On the two maps provided with the base game there are two Lime Stations, which appear as a tall tank standing on thin supports.
If you bring a machine that can contain Lime next to one of these Lime Stations, you'll get the option to refill the machine with Lime. Hit the Refill button 
Refill MachineDefault Buttons: 
R
??
?? to start filling your machine with Lime. You will pay for each liter you load. You can stop refilling prematurely by hitting the same button again. Alternatively, refilling will stop if you run out of money, or if the machine is driven away from the Lime Station.
Purchasing Lime at a Lime Station is significantly cheaper than purchasing it at the Store (see next subsection).
You can refill Fertilizer Spreaders directly from Lime Stations without the need for any other loading machines. You may then unload Unload MachineDefault Buttons: I ?? ?? the machine, causing all Lime content to be dropped in Pallet form, which is useful for long-term storage. However note that these pallets are extremely heavy.
Alternatively, you can refill a Tipper or any other valid Container with Lime as well. Lime cannot be transferred directly from such a container into a Fertilizer Spreader. Instead, it can be dumped on the ground as a Heap, and then loaded back into a machine using Loaders or Conveyor Belts. While complicated, this method allows you to store a lot of Lime on your farm in loose form and load them quickly into your machines.
Store
You can also purchase Lime at the Store. This Lime comes in Pallet (BigBag) form, containing 2,000 liters of Lime each. Lime purchased this way is significantly more expensive than at Lime Stations. The advantage is that Pallets can be transported on Bale Trailers or other similar object transporters, stacked at your farm, etc.
Note that Lime Pallets are extremely heavy compared to other 2,000 liter BigBags. A Skid Steer or Forklift can easily tip over when trying to lift one up.
If you want Lime in Pallets but don't want to pay the higher price at the Store, consider instead refilling a Fertilizer Spreader at a Lime Station and then unloading it where you want to store the Lime (see prev. subsection).