Does HP Vertica run on Windows and is it compatible with our existing Oracle SQL?
Vertica only runs on Linux, you can have a Windows client to acesso Vertica. But don't get scared Linux is not such a scary dragon kkkkk. Vertica installs as 1,2,3 check this link for a full installation article.
  Vertica uses SQL ANSI so no need for you to learn other languages, it has it's differences but is easy to work with.
 No need for you to break you brain in doing performance tuning and all those Oracle crazy queries, or create a billion indexes in SQL SERVER, Vertica does this for you. 
With the Database designer you can throw your query into Vertica and Vertica will create the optimum design for you.

I think that Vertica arch is so simple that people are confused about it !! kkkk  
HOST: is a computer where the Vertica is installed.
- is a linux box that has the Vertica software installed with the required configuration
to see the configuration needs go to this link for a full how to tutorial

INSTANCE: it is also a computer where the Vertica is in running state. 
- this is the linux box with Vertica installed and up and running(as a single node or as part of a cluster) take a look at this link for more details about Vertica arch topology

Node: it is the combination of HOST &amp; INSTANCE.
 Once more the link i gave you would explain this point. A node is a part of a cluster that has the Vertica software up and running on it. The instance is made of the processes that Vertica software creates when the Vertica engine is started.
 So in short Node is made of the Host(is the linux box with the Vertica software) and Instance(are the processes that the Vertica software creates) - this is not specifically for Vertica, it applyes to all databases.

CLUSTER: I am not clear about this. May be this is about the instances of Vertica running on different computers.
- a cluster is a collection of Vertica instances. Why do cluster is good ? MPP(massive parallel processing), HA(high availability).

DATABASE: It is the actual installed database software where the data &amp; catalog resides to which other nodes are connected (not sure abt this also).
The database is combination of Software(all those binary+ more files filed with code + datafiles(these files hold the data that is processes by your database)) and the instance (once again the instance is the process created by your database software).
- I hope i could help ! 
Feel free to visit this link  for more Vertica scripts, articles and tutorials.
Also this link