Databases can easily be put into a cloud environment as a hosting platform, but according to the NIST definition of Cloud Computing (csrc.nist.gov/publications/nistpubs/800-145/SP800-145.pdf), many existing databases cannot be easily migrated to a cloud computing model since they do not meet one of the five essential characteristics of Cloud Computing, specifically "Rapid elasticity" – since according to the NIST definition, they cannot be easily “elastically provisioned and released, in some cases automatically, to scale rapidly outward and inward commensurate with demand.”
While “database” is a general term, many databases in use today are relational databases and are designed to support transaction processing and do not natively support “rapid elasticity.” The reason for this limitation is that some essential properties of relational databases are the so-called ACID properties (Atomicity, Consistency, Isolation, Durability), which are required to support transactional processing. Some databases in the cloud (including many NoSQL databases) have tried to meet this “rapid elasticity” requirement by relaxing some of the essential transaction processing properties to make transactions “eventually consistent,” but in doing so cannot guarantee consistency.
Therefore, I believe to answer your question, we need to distinguish between “databases” and “ACID compliant databases or those databases that support transaction processing” where the “ACID compliant databases” might not quality as “cloud databases” when using the NIST definition due to their current inability to exhibit “rapid elasticity.”
A database accessible to its clients from the cloud and delivered to users on demand via the Internet from a cloud database provider's servers. This is also referred to as Database-as-a-Service (DBaaS). A cloud database can be a traditional database such as a MySQL or SQL Server database that has been adopted for cloud use, a native cloud database such as Xeround's MySQL Cloud database tends to better equipped to optimally use cloud resources and to guarantee scalability as well as availability and stability.
Cloud databases can offer significant advantages over their traditional counterparts, including increased accessibility, automatic failover and fast automated recovery from failures, automated on-the-go scaling, minimal investment and maintenance of in-house hardware, and potentially better performance. At the same time, cloud databases have their share of potential drawbacks, including security and privacy issues as well as the potential loss of or inability to access critical data in the event of a disaster or bankruptcy of the cloud database service provider.