Flash memory is used for device software and applications. It used in much the same way as RAM memory in your computer.
Expanded memory is contained on a separate microSD card. It is not directly addressable and usually used to file storage. It functions much like the Hard Disk Drive in your computer.
Finally, On-board Memory. This has only used on Bold, it appears there is 1 Gb of memory installed, of which 128 Mb is used for Flash Memory and the remainder used as Expanded memory, but on a fixed internal chip.