This question already has an answer here:
I have a set of adf file which use the same structure. How can I merge all ADF to shp file? Does the ogr2ogr command can help in this case?
The ADF file format is the Arc/Info Binary file format. Like a SHP file, the ADF file is actually several files, that together act as one file.
ADF is also a raster file, which means you cannot convert it directly to a Shapefile without, as @warrieka suggested, first converting the raster to vector shapes. Shapefile only supports vector data, not raster data.
If you are wanting the ADF raster data in one file you might consider converting it to something like GeoTIFF. GeoTIFF is widely compatible and multipurpose. But you can convert it to anything you would like or suits your needs the best.