How do I tell if it's really Ivory?

There are NO definitive, simple, single, casual tests to definitively identify ivory.  However, with some work, someone with some training and good tools can generally tell elephant ivory from all other ivories and from the imitations. 

The forensic process consists of building up a body of evidence that supports the use of ivory. 

The most definitive structure in ivory are the Rings of Shraeger as shown in this picture.

The Convention on the International Trade in Endangered species provides a booklet available in pdf format which tells you how to judge the angles in the Rings of Shraeger to see what animal it comes from.

If you see the Rings of Shraeger in the projecting mounts on a bagpipe, you probably have elephant ivory, but you really should confirm it with another test as there is an imitation ivory that can easily fool a careless observer unless they take the time measure the angles. 

Some people suggest a hot needle/smell test to tell between ivory and non-ivory, but unless you are willing to ruin your piece and you know what you are smelling for, it doesn't work and isn't definitive.  Some people will claim that a black light will make real ivory glow.  This is also not a definitive test for ivory.

A shop that does repairs can generally make an excellent assessment for you.

(Before you ask - I only identify ivory for myself.  When it's my $$ on the line, I rely on a combination of microscopy, fluorescence and infra-red spectroscopy to identify the material in question.  These tools can reveal a great deal about the identification of non-elephant ivory or ivory substitutes as well.)