I have some marked "Secretarial" and some marked "88" and none with both. Some of them take regular standard "universal" spools and other take the Smith-Corona spools with the very large opening in the bottom to fit over the reversal mechanism, and that's the distinction, that large hole. If your spools have that, you need the S-C spools or you should rewind any ol' ribbon onto your original spools and you'll be fine. If your spools have the same size spindle hole on both sides, you can use  a universal spool.

In any case, don't discard the metal spools. People want them. 

A further clarification -- if the typewriter has the special Smith Corona spools (and I think it does), then those don't work with the "universal"-- type ribbon that has metal grommets at each end, unless you perform a bit of surgery on the ribbon. You'll need to modify the ribbon to cut off the ends, including the grommets. The grommets will interfere with the ribbon reversing mechanism. 

Yes you definitely have the large spools and you will have to rewind the fabric from a 'universal' ribbon.  If that ribbon has metal eyelets set into the ends, you must cut that section of ribbon off to get rid of the eyelet.  Rewinding is actually pretty simple. Remove the old ribbon from the machine, run the old fabric off into a wastepaper basket then unhook the ends from the little metal arrow-heads.  Take your new ribbon and detach the end from one of the spools (they usually come with two plastic spools these days)  Hook the loose end onto the arrow head on the metal spool.  There are different methods for winding the fabric, but an easy way is to roll the spool along a flat surface, following behind with the plastic spool paying the fabric out.  Once the metal spool is full, unhook and discard the remaining plastic spool and hook the end of the fabric onto the second metal spool.  All done

Just to follow on to thetypewriterman's description, make sure you're winding on the ribbon in the correct direction, since if you mess up you can't merely flip the spools as you can with the universal spools. I'm pretty sure (but confirm!) that the right spool winds on ribbon counterclockwise, while the left spool winds on ribbon clockwise. There may even be arrows stamped into the spool cups on the machine showing this, or just play with the machine with no ribbon spools mounted to make sure..

,<giving myself a dope slap> Of course you are correct. Sorry to be an alarmist.