Both Flash and Shockwave are multimedia players. They can give you extended and predictable abilities across a range of browser brands, versions, and platforms. (Sometimes you might hear someone refer to "Shockwave Flash", but these are actually two different multimedia players.)
Flash has a small player which gives it a wider distribution. Flash is included in every Netscape download. Flash also has a very fast startup time. The way the Flash format interleaves media and instructions also helps it start quickly.
Shockwave has a deeper player. It offers multiuser chat, XML parsing, HTML manipulation, an extensive and fast scripting language, distant file retrieval, programmatic control of vector shapes, and bitmap manipulation. It can do many amazing things which browsers cannot do. Its files also stream, but it does not start instantly like Flash does.
Both are ubiquitous. They are distributed in all popular operating systems, browser CDs, and installed on new computers. Flash files can play on more platforms, in large part because its file format is published.
The Shockwave file format is .DCR, and it is created by the Director authoring application. The Flash file format is .SWF, and is created by Flash, FreeHand, Generator, and other tools.
Use Flash for interfaces to sites, for quick impact. Use Shockwave for more complex multimedia work, or for web applications that are beyond the browsers' abilities.