This article explains the different file formats that use the DST file extension. It also explains how to open each type, and what options you have for converting your specific file to a different format.
A file with the extension .DST can be a character set created by Autodesk's AutoCAD. It contains various drawing layouts (in XML format) taken from DWG files.
Tajima Embroidery Format also uses the DST file extension. The file stores stitch information that describes how the software should control the sewing needle. It is used by various embroidery machines and programs.
Other DST files may be DeSmuME save state files associated with the Nintendo DS emulator DeSmuME. These are created when you save the game state in that software.