Piling is the general process in which building foundations are laid. They provide stability and security to the building and is the glue that holds it together. Piling can also be rather expensive but relatively speaking not so. What I mean by this is that piling costs a fair amount but without it a house cannot be constructed which is a lot more expensive than piling!
Mini Piling is basically piling except, you know, mini. It is most commonly used when it comes to conservatories, extensions and orangeries. This is because these are small (again relatively speaking) and more maneuverability is required. This is why mini piling is preferred in these situations.
Underpinning is basically piling except with a different material. This means the tubes and work is not made out of metal allowing it to be a lot cheaper. This is perfect for small extensions, conservatories or orangeries because their small weight (again again relatively speaking) can be supported by the underpinning.