Apricorns can be found on little trees in any forest biome. Right-click a bush to get the visible apricorn on top (in version 1.4.6 and lower it's a little bit buggy and you have to click a few times when the tree is highlighted with a thin black block border), and if you left click it will destroy the bush giving you an oak wood log in the process. Apricorns can be replanted on grass next to any other blocks and grown to harvest more apricorns. Apricorn trees do not require a source of water to grow, unlike Minecraft's other plants.
Apricorns are used to make Poké Ball discs, and in order to make discs you must cook the apricorns in a furnace. Putting three red cooked apricorns in a row in a crafting bench will make a normal Poké Ball disc. All discs are made with color patterns in a row in a crafting bench. The discs can then be used with an Anvil and a Hammer to make the corresponding Poké Ball Lid.