To be quite sure, you need some equipment including a good magnifying glass, needles or pins and a flame - you can find the full list of recommended equipment here.

Real ivory, when looked at closely is not smooth but has lines or grain patterns running across it. These are the lines that run across the tooth from which the ivory was carved. Elephant or mammoth ivory has V shaped or circular lines on it. (Mammoth ivory is starting to appear more now; because there is very little trade in elephant ivory, in some places such as parts of China people have been searching for mammoth remains instead).

If there are no lines, you may be looking at a piece of bone rather than ivory. Bone has grooves or canals that allow nutrients to pass through, which teeth don't. There may also be dark patches on bone if you look closely.

A full guide to testing for ivory can be found here.