Borehole leaching (also known as in-situ leaching) consists of using boreholes in the ground the extract minerals from the surrounding rock. The process involves drilling boreholes and using explosives or hydraulic fracturing to create gaps in the surrounding rock. Then chemical leaching solution are pumped down the borehole and into the gaps. This solution becomes saturated with the target minerals and is then pumped back up to the surface for processing.
Heap leaching on the other hand is not done in-situ. Instead, broken ore is piled into "heaps" or "dumps" and then leached. Heap leaching is increasingly used in the recovery of gold, silver and copper.
The "pregnant" leach solution from either borehole or heap leach is collected and then processed to extract the minerals.
In both cases, environmental precautions must be taken to avoid interaction between leaching solutions and water resources.
Figure: Above is a simple processing diagram of using heap leaching to process metals. A similar process could be used for borehole leaching solutions.