What is al-Jarh wa al-Ta'dil in the Prophetic Hadith, and how and when is it applied, with real-world examples?

The science of al-Jarh wa al-Ta'dil (discrediting and accrediting) is concerned with studying the conditions for accepting narrators of hadith and the reasons for rejecting them. Whoever meets the conditions is accepted, and whoever does not meet them is rejected. An example of an accepted hadith is what al-Bukhari reported from Ibn Umar concerning the parable of the Ummah. An example of a rejected hadith is what Ahmad reported from Abu Hurairah regarding the sacrifice, where there is a narrator in its chain of transmission who was not deemed trustworthy.