Why is the consensus problem so important in distributed computing?
The paper you mention is important for 2 reasons:


It shows that there is no asynchronous deterministic consensus algorithm that tolerates even a single crash fault. Note that in the synchronous setting,there is a deterministic algorithm that terminates in $f+1$ rounds when $\le f$ processes crash.
It introduces bivalence and univalence of configurations (*), which are used in many lower bounds and impossibility proofs later on. 


Applications

One important application of the consensus problem is the election of a coordinator or leader in a fault-tolerant environment for initiating some global action. A consensus algorithm allows you to do this on-the-fly, without fixing a "supernode" in advance (which would introduce a single point of failure). 

Another application is maintaining consistency in a distributed network: Suppose that you have different sensor nodes monitoring the same environment. In the case where some of these sensor nodes crash (or even start sending corrupted data due to a hardware fault), a consensus protocol ensures robustness against such faults.



(*) A run of a distributed algorithm is a sequence of configurations. A configuration is a vector of the local states of the processes. Each process executes a deterministic state machine. Any correct consensus algorithm must eventually reach a configuration where every process has decided (irrevocably) on the same input value. A configuration $C$ is $1$-valent if, no matter what the adversary does, all possible extensions of $C$ lead to a decision value of $1$. Analogously, we can define $0$-valency. A configuration $C$ is bivalent if both decisions are reachable from $C$ (which one of the two is reached depends on the adversary). Clearly, no process can have decided in a bivalent configuration $C$, as otherwise we get a contradiction to agreement! So if we can construct an infinite sequence of such bivalent configurations, we have shown that there is no consensus algorithm in this setting.