The story of database persistence in Java is already long and full of twists and turns:
JDBC is the low level API that everybody uses at the end to talk to a database. But without using a higher level API, you have to do all the grunt work yourself (writing SQL queries, mapping results to objects, etc).
EJB 1.0 CMP Entity Beans was a first try for a higher level API and has been successfully adopted by the big Java EE providers (BEA, IBM) but not by users. Entity Beans were too complex and had too much overhead (understand, poor performance). FAIL!
EJB 2.0 CMP tried to reduce some of the complexity of Entity Beans with the introduction of local interfaces, but the majority of the complexity remained. EJB 2.0 also lacked portability (because the object-relational mapping were not part of the spec and the deployment descriptor were thus proprietary). FAIL!
Then came JDO which is a datastore agnostic standard for object persistence (can be used with RDBMS, OODBMS, XML, Excel, LDAP). But, while there are several open-source implementations and while JDO has been adopted by small independent vendors (mostly OODBMS vendors hoping that JDO users would later switch from their RDBMS datastore to an OODBMS - but this obviously never happened), it failed at being adopted by big Java EE players and users (because of weaving which was a pain at development time and scaring some customers, of a weird query API, of being actually too abstract). So, while the standard itself is not dead, I consider it as a failure. FAIL!
And indeed, despite the existence of two standards, proprietary APIs like Toplink, an old player, or Hibernate have been preferred by users over EJB CMP and JDO for object to relational database persistence (competition between standards, unclear positioning of JDO, earlier failure of CMP and bad marketing have a part of responsibility in this I believe) and Hibernate actually became the de facto standard in this field (it's a great open source framework). SUCCESS!
Then Sun realized they had to simplify things (and more generally the whole Java EE) and they did it in Java EE 5 with JPA, the Java Persistence API, which is part of EJB 3.0 and is the new standard for object to relational database persistence. JPA unifies EJB 2 CMP, JDO, Hibernate, and TopLink APIs / products and seems to succeed where EJB CMP and JDO failed (ease of use and adoption). SUCCESS!
To summarize, Java's standard for database persistence is JPA and should be preferred over others proprietary APIs (using Hibernate's implementation of JPA is fine but use JPA API) unless an ORM is not what you need. It provides a higher level API than JDBC and is meant to save you a lot of manual work (this is simplified but that's the idea).