Consensus algorithms are important for all kinds of distributed computing problems. A simple example would be failover. If you have a leader database that replicates to 4 others, and you want another node to take over if the leader DB fails, then you need a consensus algorithm to prevent a situation where 2 different machines both think they're the new leader in a netsplit.

There are many other equivalent problems in distributed computing, from atomic transactions to "exactly once" messaging systems.

Asynchronous consensus is a model where you cannot make assumptions about the bounded nature of call timings, whereas in a synchronous model, you can assume everything is bounded.

Byzantine fault tolerance is important for security under byzantine faults, that is to say malicious actors acting deliberately against what the protocol specifies they should do.