@Callithumpian has given a nice representation of the words usage in America.
What are the actual sources of these words, and what words are used in other English-speaking areas, such as Australia and South Africa?
All of these words actually just refer to the "sports shoe" or "atheletic shoe"
Here are their origins:
The British English term "trainer" derives from "training shoe." There is evidence that this usage of "trainer" originated as a genericized tradename for a make of training shoe made in 1968 by Gola...
This is "sneaker":
In the 1800s, a London police officer developed a rubber-soled shoe in order to catch criminals in the act quietly. He called his invention "sneakers". The name derived from the fact that the rubber soles of the shoes made them noiseless. The term "sneaker" was also used in 1887 by Boston Journal of Education:
“It is only the harassed schoolmaster who can fully appreciate the pertinency of the name boys give to tennis shoes — sneakers.”
In other countries, they're also called:
They are also known as trainers (British English), sandshoes, gym boots or joggers (Australian English), running shoes, runners or gutties (Canadian English, Australian English, Hiberno-English), sneakers, tennis shoes (North American English, Australian English), gym shoes, tennies, sports shoes, sneaks, tackies (South African English and Hiberno-English), rubber shoes (Philippine English) or canvers (Nigerian English).