In a nutshell, ZooKeeper helps you build distributed applications.
You may describe ZooKeeper as a replicated synchronization service with eventual consistency. It is robust, since the persisted data is distributed between multiple nodes (this set of nodes is called an "ensemble") and one client connects to any of them (i.e., a specific "server"), migrating if one node fails; as long as a strict majority of nodes are working, the ensemble of ZooKeeper nodes is alive. In particular, a master node is dynamically chosen by consensus within the ensemble; if the master node fails, the role of master migrates to another node.
The master is the authority for writes: in this way writes can be guaranteed to be persisted in-order, i.e., writes are linear. Each time a client writes to the ensemble, a majority of nodes persist the information: these nodes include the server for the client, and obviously the master. This means that each write makes the server up-to-date with the master. It also means, however, that you cannot have concurrent writes.
The guarantee of linear writes is the reason for the fact that ZooKeeper does not perform well for write-dominant workloads. In particular, it should not be used for interchange of large data, such as media. As long as your communication involves shared data, ZooKeeper helps you. When data could be written concurrently, ZooKeeper actually gets in the way, because it imposes a strict ordering of operations even if not strictly necessary from the perspective of the writers. Its ideal use is for coordination, where messages are exchanged between the clients.
This is where ZooKeeper excels: reads are concurrent since they are served by the specific server that the client connects to. However, this is also the reason for the eventual consistency: the "view" of a client may be outdated, since the master updates the corresponding server with a bounded but undefined delay.
The replicated database of ZooKeeper comprises a tree of znodes, which are entities roughly representing file system nodes (think of them as directories). Each znode may be enriched by a byte array, which stores data. Also, each znode may have other znodes under it, practically forming an internal directory system.
Interestingly, the name of a znode can be sequential, meaning that the name the client provides when creating the znode is only a prefix: the full name is also given by a sequential number chosen by the ensemble. This is useful, for example, for synchronization purposes: if multiple clients want to get a lock on a resource, they can each concurrently create a sequential znode on a location: whoever gets the lowest number is entitled to the lock.
Also, a znode may be ephemeral: this means that it is destroyed as soon as the client that created it disconnects. This is mainly useful in order to know when a client fails, which may be relevant when the client itself has responsibilities that should be taken by a new client. Taking the example of the lock, as soon as the client having the lock disconnects, the other clients can check whether they are entitled to the lock.
The example related to client disconnection may be problematic if we needed to periodically poll the state of znodes. Fortunately, ZooKeeper offers an event system where a watch can be set on a znode. These watches may be set to trigger an event if the znode is specifically changed or removed or new children are created under it. This is clearly useful in combination with the sequential and ephemeral options for znodes.
A canonical example of Zookeeper usage is distributed-memory computation, where some data is shared between client nodes and must be accessed/updated in a very careful way to account for synchronization.
ZooKeeper offers the library to construct your synchronization primitives, while the ability to run a distributed server avoids the single-point-of-failure issue you have when using a centralized (broker-like) message repository.
ZooKeeper is feature-light, meaning that mechanisms such as leader election, locks, barriers, etc. are not already present, but can be written above the ZooKeeper primitives. If the C/Java API is too unwieldy for your purposes, you should rely on libraries built on ZooKeeper such as cages and especially curator.
Official documentation apart, which is pretty good, I suggest to read Chapter 14 of Hadoop: The Definitive Guide which has ~35 pages explaining essentially what ZooKeeper does, followed by an example of a configuration service.