If you have RDBMS you probably have to use Solr to index your relational tables to fully nested documents.
Im new to non-sql databases like Mongodb, CouchDB and Cassandra, but it seems to me that the data you save is already in that document structure like the documents saved in Solr/Lucene.
Does this mean that you don't have to use Solr/Lucene when using these databases?
Is it already indexed so that you can do full-text search?
It depends on your needs. They have a full text search. In CouchDB the search is Lucene (same as solr). Unfortunately, this is just a full text index, if you need complex scoring or DisMax type searching, you'll likely want the added capabilities of an independent Solr Index.
Solr (Lucene) uses an algorithm to returns relevant documents from a query. It will returns a score to indicate how relevant each document is related to the query.
It is different than what a database (relational or not) does, which is returning results that matches or not a query.