Typically some combination of 1) sneezing 2) mucous discharge from or stuffing in the nose 3) mucous or watery discharge from the eyes, and it can also include 4) fever and if it progresses, 5) lethargy and 6) loss of appetite, and 7) coughing (if it gets into the lungs). With one particular URI virus (calici) you can also get sores in the mouth and joint pain, but this is not a common URI.
If your cat has any of these symptoms you will want a vet visit so they can put the cat on meds to prevent a secondary bacterial infection which can be worse than the original URI. You may also need to do some things to keep his or her appetite up. If a cat stops eating during a URI (often if they get stuffed up and can't smell their food, then they won't eat it) then the URI become more serious.