A month ago I bought domain, petz-dating.com. Today while I was surfing I came across a page, http://patest.petz-reloaded.com, where it said "Petz is a trademark, and Catz and Dogz are registered trademarks of Ubi Soft Entertainment, Inc." What does mean trademark? Should I change domain name?
Maybe. http://www.uspto.gov/main/trademarks.htm Looks like it might be in the process if these are the same people. Otherwise you both may be in the same spot. Trademark means someone has paid the fees and registered the name as "mark" of their business.
Trademarks are granted by classification - and there are over 40 different classifications. If your usage does not conflict with an existing mark and its classification, you should be fine. For instance, there are several owners of a trademark on "Apple" or "MySpace" - each having different usage rights. Also, doing a check on the USPTO.gov site is only the first step - In the US, a company does not have to register their mark to establish usage rights.