If it's an HTML-only website, then you can simply upload the same files to a subdomain like test.domain.com
For WordPress conversions, there are thee ways to test it:
Test it locally on something like WAMP. We use this for quality control ourselves.
Host it on another domain or IP address. Then change your "hosts file" in windows to tell your browser that the domain is hosted on a different IP. Here is a Windows guide and a guide for Mac/Linux users.
Pay us an extra $20. Then we make the files compatible for another domain, so you can test it on another (sub)domain. Please then specify which domain you want to use for testing, for example, test.domain.com.