Lucene is a search library written in Java. ElasticSearch is a server build around the Lucene library and allows you to interact with Lucene through an easy to use HTTP interface.
You can use Lucene directly in your application but you will need to cover a lot of integration ground work. By using ElasticSearch (or Solr for that matter) a lot of the integration work is taken care of and you can use the search technology in a decoupled manner since the search aspect will run in an independent server environment.