A "cold"... ... Is a viral infection. Allergic noses drip and itch, viral noses have a more "gooey" mucus. Cough and fever are more likely with a viral infection; tearing eyes are more likely with an allergy.
Length of symptoms. This can be difficult to tell. Allergies can resemble colds. However, some clues are length of symptoms (cold typically last 7 to 14 days). Whether symptoms correlate with allergen exposure is another clue. Also, allergies do not cause fevers. Unfortunately the color of mucus discharge is not a reliable predictor. Skin or blood allergy testing can be helpful to make this determination.
Fever or No fever. Nasal and ocular allergies feel just like a cold--stuffy, runny noses, scratchy or itchy throats, sneezing, itchy eyes, but usually no fever, and little fatigue (some people say they feel "foggy" and tired, but not fatigued). Viral infections, aka "colds" are usually accompanied by fever or increased temperature, sweating, congestion, runny nose, sore throats, etc.
Fever or aches. Both colds and allergies can give you a runny/stuffy nose and coughing, but if there are aches or a fever it's a cold (though not all colds will cause fever and aches). Colds tend to last less than 7 to 10 days, allergies can last weeks to months - or if it's an acute exposure to something your allergic to, it may only last minutes.
Good question. Some time it is hard to tell. In general, when your child has a cold, someone else in the family often has one. A cold is often associated with a low-grade fever but allergy is not. You may also notice fatigue as well as joint or muscle ache with a cold but not with allergy. Hay fever may respond to antihistamine but a cold does not.