Up to version 8, Adobe Illustrator’s native format was based on PostScript (1.5.1.1 Adobe’s Vector Formats). It was not fully standard PostScript, however, and importing it into other software has always been a pain. There now exist various scripts on the Web which claim to convert this old AI format into something more tractable, but they are all rather limited and not too reliable. While you can try to use them, officially Inkscape 0.47 does not support import or export of the old AI format.
Starting from version 9, Adobe switched to PDF for the base of its new Illustrator format. The AI files saved by Illustrator still contain many AI-specific extensions; however, they are standard PDF and any PDF-capable software can open and view them as PDF. This is what Inkscape does: It treats any file with the .ai extension as a PDF file and presents its standard PDF Import Settings dialog to you (B.3 PDF (Import, Export)). Inkscape will thus lose the AI-specific metadata (such as layers), but at least you will get your vector objects as vectors.
One feature used in complex AI files is gradient meshes (1.5.4 . . . and Inkscape). Inkscape can import them, but since there’s no similar construct in SVG, it has to approximate them with a lattice of small flat-color path “tiles.” In the PDF Import Settings dialog, you can set the precision of this approximation; raising this parameter will make the imported mesh look smoother, but at the cost of increasing the size of the SVG file and slowing down Inkscape.
Imported gradient meshes, with their lattices of small colored paths, are a convenient object for moving, painting, and reshaping in the Tweak tool (6.9 Transforming with the Tweak Tool, 8.7 Color Tweaking, 12.6 Path Tweaking).
There’s no support for AI export in Inkscape, because all recent versions of Adobe Illustrator can import SVG and PDF files without problems.