When you save a file using a certain program, the file is often saved in a proprietary format only that program can recognize. For example, if you save a Microsoft Word document, it is saved as a Word document (i.e. mydocument.doc). This is a native Word file -- that is, the file format is native to the Microsoft Word application and may not be recognized by other programs.
When you use the "Save As..." command to save a file, you may be given the option to save the file in a different format. For example, you might be able to save a Word document as a plain text (.txt) file or a rich text (.rtf) file. These formats are not native to Microsoft Word, but can still be opened by the Microsoft Word program. Similarly, Adobe Photoshop saves files as Photoshop documents (.psd files), but can also save them in .jpg and .gif formats, among others.
It is usually best to save a file in a program's native file format because you can be sure it will store all the data you have created with the program. While other formats may be more compatible with other programs, they might not save all the information in the file. For example, if you save a Word document as a plain text file, all the text formatting you had added will be removed. Saving a Photoshop document in JPEG format will reduce the image quality and flatten all the image's layers.
So if you're planning on editing a file after you've saved it, saving it as a native file is the best choice. If you want to save the most compatible version of the file, saving a copy of the file in a more generic format is a good idea.