The conditions of having stones in the bladder is cystolithiasis, and bladder stones are cystoliths.
The condition of having kidney stones is called nephrolithiasis
The condition of having stones in the ureters: ureterolithiasis
Anything past that, we usually do not name, aside from calling it "renal colic" if it is painful.
If you are asking for specific conditions that can lead to the formation of kidney stones, there are many. Some of these include gout, or hyperuricemia (high uric acid levels in the blood). Many medications predispose one to making stones. High calcium levels, or a condition in which the kidneys excrete a lot of calcium (hypercalciuria) also is a big risk factor. Any abnormality of the kidney also makes one more likely to make stones. Recurrent infections with certain kinds of bacteria (proteus mirabilis, especially) is a big risk factor for a certain type of stone. There are many, many more conditions that can predispose to stones.
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Cystectomy is a medical term for surgical removal of all or part of the urinary bladder.
Lithotomy is the medical term meaning surgical removal of a stone.
Lithotomy is the medical term meaning surgical incision to remove a stone.
Uterterolithotomy is surgical incision into a ureter to remove a stone.
Ureterolithotomy is surgical incision into the ureter to remove a stone.