I'm looking at a different websites like http://www.domainavailabilitycheck.com/ and I would like to know what is the source of data. Do they use whois information or is there a public API for that?
Those websites most likely use a standard WHOIS command as defined in RFC3912. This is in fact a public api the domain registrars provide. Please note that some of those websites are known for registering likely to be used domains to squat them.