On the iPhone and iPad, Safari is the standard browser. To not record a browser history, you can just stay in Private mode while surfing. When you do have a history to delete, go to Settings > Safari > Clear History & Website Data. Doing this not only takes out the history, but also cookies and other stuff. Plus, if the phone is signed into iCloud, it clears the history on iCloud as well as on other devices hooked into that iCloud account.
If you want to only delete data for select sites, go back to Settings > Safari and scroll down to Advanced > Website Data. After it loads (it can take a while) you'll see a listing of every website you've visited—and probably a lot you didn't, because it also records the sites serving third-party cookies. Click edit > minus symbol next to each to delete, or just swipe left for the same function.