Koi are really just colored common carp( cyprinus carpio) bred for many years to bring out different patterns. If you look under their mouths they have barbels, whiskers if you like, that they use to find food in the mud at the bottom of the pond. Also their mouths are lower down than goldfish and have a more humped back, but if they are small this will be hard to see.
Goldfish on the other hand come from a small brown Asian carp (carassius auratus auratus) which like the Koi has been selectively bred. Goldfish are bred for their colors similar to Koi, but less for patterns. I take it your fish are common Goldfish as they are also bred for different body & fin shapes. If they have fan-tails then they are definitely Goldfish.not Koi.
To put it simply Koi are generally multi-colored with barbels.
Goldfish are generally single colored but not always gold without barbels.
at 4:44 PM on Jun. 28, 2010