Delete all documents from index/type without deleting type

I believe if you combine the delete by query with a match all it should do what you are looking for, something like this (using your example):

curl -XDELETE 'http://localhost:9200/twitter/tweet/_query' -d '{
    "query" : { 
        "match_all" : {}
    }
}'

Or you could just delete the type:

curl -XDELETE http://localhost:9200/twitter/tweet

Note: XDELETE is deprecated for later versions of ElasticSearch

The Delete-By-Query plugin has been removed in favor of a new Delete By Query API implementation in core. Read here

curl -XPOST 'localhost:9200/twitter/tweet/_delete_by_query?conflicts=proceed&pretty' -H 'Content-Type: application/json' -d'
{
    "query": {
        "match_all": {}
    }
}'

From ElasticSearch 5.x, delete_by_query API is there by default

POST: http://localhost:9200/index/type/_delete_by_query

{
    "query": { 
        "match_all": {}
    }
}

You can delete documents from type with following query:

POST /index/type/_delete_by_query
{
    "query" : { 
        "match_all" : {}
    }
}

I tested this query in Kibana and Elastic 5.5.2

Torsten Engelbrecht's comment in John Petrones answer expanded:

curl -XDELETE 'http://localhost:9200/twitter/tweet/_query' -d 
  '{
      "query": 
      {
          "match_all": {}
      }
   }'

(I did not want to edit John's reply, since it got upvotes and is set as answer, and I might have introduced an error)