You should encrypt data at rest and in motion. Encrypting “in motion” is already well known to you – the standards of HTTPS/SSL and IPSEC apply equally well in the data center and in the cloud.
Encrypting “at rest” means that the data must be encrypted when it resides on a disk, in a database, on a file system, in storage, and of course if it is backed up. In the real world, people have not always done this in data centers – often relying on physical security as a replacement. In the cloud, physical security is no alternative – you must encrypt sensitive data.
This actually means data must be encrypted constantly as it is being written, and decrypted only when it is going to be used (i.e. just before a specific calculation, and only in memory). Standards such as Advanced Encryption Standard (AES) are commonly used for data encryption at rest.