The primary vector format used by Inkscape is called Inkscape SVG; as explained in 1.4 A Brief History of SVG, it is (almost) standard-compliant SVG with some Inkscape-specific extensions that affect only the editability of various object types in Inkscape, but never change their appearance. It does not make much sense to save as Plain SVG except when you are trying to achieve a modest file size gain or encounter some buggy software having problems with Inkscape SVG. Both SVG flavors have compressed varieties (using the .svgz file name extension), which produce much smaller files but are otherwise the same and should be understood by most SVG software.
SVG files exported from Adobe Illustrator can be opened in Inkscape as usual. However, these files usually contain a lot of AI-specific stuff, which is useless for Inkscape and just blows up the size of the SVG file. It is recommended to save SVG files from AI with the extension .ai.svg instead of simply .svg. When you open such a .ai.svg file in Inkscape, it goes through a special filter which removes the AI-specific binary chunks and converts AI layers into Inkscape layers.