All users of cryptocurrencies see them every day.
Each transaction gets into a Blockchain. During the mining, different people have different chain of transactions. Before the longest chain is implemented, it could be said there is a fork in a chain.
The most significant example for applying UASF and UAHF is a history of Ethereum.
In April-May, 2006, Ethereum started The DAO project. But a month later a hackergot the money of holders out. The sum was about $55,000,000. After the incident three solutions were suggested:
to accept the theft and do nothing
to rollback the Blockchain to time before the theft, or hard fork
to accept all the transactions to the hacker’s wallets false and return the money back, or soft fork
The Ethereum community chose to go with the second variant. But some users weren’t happy with that. They assumed that “code is law” is a main concept of cryptocurrency. These users accepted the theft, so the second Blockchain appeared. The first chain is known as Ethereum, ETH, the second chain is known as Ethereum Classic, ETC.