Dolphin is an emulator for playing Wii and GameCube games. While the site is safe, and the download is as well, emulators are sometimes considered illegal. It's completely safe if you make dumps for your games yourself, though.
This is the official website of Dolphin emulator. It is also affiliated with the official wiki, and is also the site linked to by Wikipedia and journalists alike.
Do not go to the .com site; it is a scam.