Ceramic tile is the most common and effective floor covering for radiant floor heating, because it conducts heat well and adds thermal storage. Common floor coverings like vinyl and linoleum sheet goods, carpeting, or wood can also be used, but any covering that insulates the floor from the room will decrease the efficiency of the system.
If you want carpeting, use a thin carpet with dense padding and install as little carpeting as possible. If some rooms, but not all, will have a floor covering, then those rooms should have a separate tubing loop to make the system heat these spaces more efficiently. This is because the water flowing under the covered floor will need to be hotter to compensate for the floor covering. Wood flooring should be laminated wood flooring instead of solid wood to reduce the possibility of the wood shrinking and cracking from the drying effects of the heat.