Partitioning of Data

This involves moving different tables or different parts of the database into different databases, and normally on different servers. This allows you to scale an application by adding more database servers, so your data could be spread across several servers.

HyperDB allows you to specify which data resides on which servers.
Replication

Replication involves having one database server act as the master (the central copy of all data) and other servers act as slaves (each with a copy of the data). All the database writes must go to the master server, but database reads can be sent to slave servers, thus spreading the load amongst multiple servers.

HyperDB allows you to split database writes and reads and assign each to an array of servers.
Failover

HyperDB allows multiple servers to be configured and multiple routes to servers. If the first server can't be accessed the second server is tried, if that's down the third server, and so on. This allows for failover from internal nics to public nics to remote datacentres, etc.