Determining the age of a goldfish is not an easy task. Some signs to take into consideration are the variety, coloration, appearance and size of the goldfish in question.
When determining the age of a goldfish, it is important to know which variety is in question. Straight tail goldfish like commons, comets, sarasa commons and shubunkins develop differently than fantail goldfish like red fantails, calico fantails and black moors.
Observe the coloration of your goldfish. If it is bronze in color it is still very young. Juvenile goldfish are bronze in color and as they mature, your goldfish colors will change. Commons or comets will develop metallic orange coloration, shubunkins will become calico patterned, black moors will develop into a velvet black in color, etc.
Observe the appearance of your goldfish. The caudal fin (tail fin) of an adult goldfish will be more pronounced than a juvenile goldfish. Younger Commons, Comets, Sarasa Commons and Shubunkins will have a more rounded caudal fin and as they age, their caudal fin will become more sharply forked. Younger Red Fantails, Calico Fantails and Black Moors caudal fins are shorter and as they age their tails will grow in length, becoming more prominent. Also, adult goldfish will develop "salt spots" on their operculum (gill cover) and pectoral fins.
Lastly, the size of your goldfish will help determine its age. Mature goldfish can reach 12" or more in length. If your goldfish is considerably smaller, it is younger in age or its growth has been stunted.