Which country executes the most would be hard to say, for sure ... Since many countries ruled by tyrants or corrupt governments execute thousands of people a year - but most of them are listed as "missing" since the government doesn't actually admit to killing them.
The US has by far the largest number of prisons (4575); as well as the most prison inmates (2.2 million)---unless you count North Korea, which IS a prison of 25 million inhabitants.
Saudi Arabia is currently executing more people as a function of its penal system than any other sovereign country. Other entities such as Boko Haram and ISIL may exceed the Saudi total of executions.
I answered a similar question some months ago. China has the greatest number of executions. Nobody knows how many but they number in the thousands. The US comes in about fourth (possibly fifth) on the list.
I don't recall who has the highest number of prisoners although I suspect that China again leads the field, with the mass imprisonment of Uighurs and religious dissidents alike. Putin's Russia may well be in second place as he tries to force his brand of communism on their struggling economy.