Jacek is spot on with pointing you at those two threads which look to answer your question. The book goes through a few different modelling scenarios to try and give a feel for this which includes an ACL (access control) type scenario, modelling entities from a physically connected network such as stations on the underground, as well as a band modelling example where you may want to keep track of which musicians played which instruments on recordings by various rock bands. These are but a few, but essentially anything where you want to query highly connected data is generally a good fit.
We also point you at the general Neo4j website as well which describes some high level scenarios where Neo4j has done well, available here http://neo4j.com/use-cases.
Additionally the following website provides a collection of various people who have used Neo4j for all sorts of different problems, and are happy to share their examples with all: http://gist.neo4j.org/
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