A hiccup is a jerky, repeated contraction in your diaphragm — it's a sort of involuntary tic you can feel in your throat and chest. People who have hiccups usually make a "hic" sound.
Hiccups are a kind of spasm that can be brief or linger frustratingly. One man famously suffered from hiccups for 68 years — for that whole time, not a single folk remedy (doing a headstand, drinking water fast, being scared by someone, and so on) worked to relieve them. The word hiccup is imitative of the sound of hiccups. An Old English word for it was ælfsogoða, "elf heartburn," since hiccups were thought to be caused by elves.