Assuming you've created a schematic in EESCHEMA and generated a net list and read that into PCBNEW (in other words, you're not making a board w/o a schematic), the parts will have a proper reference and a value. Both types can be made visible and invisible. Hover over the reference (U1, etc.), press E, if a choice menu appears, select the desired reference text, a form should appear (footprint text properties). There you should see that one can choose invisible or visible. Note: A part put on the board using the O command in PCBNEW will still have the reference, but it not read as U1, R1, etcetera; it will say whatever was entered when the part was made (e.g., SOIC-8***).
If invisible was selected by default, one can edit the part globally to set the reference to always visible. There may be a way to force all references as visible, but I'm not sure of that...