A rather hard to find ingredient that is used in some Caribbean cocktails, perhaps most notably in the Bermuda Rum Swizzle. It is essentially a sweet syrup that is flavored with lime juice, ginger, and other spices. While you may sometimes see grenadine recommended as a substitute, it really isn't at all the same. For a long time, Falernum had been made by Sazerac Co., but due to apparent lack of sales, they stopped producing it. To help fill in the gap, Da Vinci Gourmet (a flavored syrup company) started producing a non-alcoholic version that could be used in cocktails that required Falernum. "Fee Brothers" as well as various other cocktail syrup companies are also now producing non-alcoholic Falernum syrups as well. If you are lucky you may be able to find a brand called "Velvet Falernum" which is made on the island of Barbados by the R. L. Seale company, and I think is the only brand that still contains alcohol.

The following recipes on this site use falernum:

Bermuda Rum Swizzle
rum, lime juice, pineapple juice, orange juice, falernum,
Chartreuse Swizzle
Chartreuse, falernum, pineapple juice, lime juice,
Jolly Roger
rum, rum, orange juice, falernum, Angostura Bitters,
rum, Benedictine, falernum, lime juice, Angostura Bitters,