OK so spatial join by location should work then, right click your building layer and choose joins and relates > join. Use the drop down at the top of the dialogue to change from an attribute join to a spatial location join. Then select the layer you want to join them to and select the second radio button down to make sure each polygon is given all of the attributes of the polygon it falls inside, as in the attachment.
Once you've done this you'll have a feature class or shapefile with the attributes of both polygons, you can then sum the area of the buildings per polygon by doing a dissolve based on the polygon ID within your building layer and divide this figure by the area of the flood map polygons. Times this figure by 100 to work out the percentage of floodplain covered by buildings. You can do this by adding a field in the attribute table to do your calculations in or you could export the tables and do it in excel, up to you I guess.
Hope that helps.