The Electronic Frontier Foundation offers handy step-by-step directions for erasing your search history here. And you can click here for instructions on deleting your YouTube search and viewing history.
Google has a page that explains how to clear the history from its various products.
Why might you want to? Maybe you don't. Perhaps you don't mind that Google just wants to cash in on all the data it's amassing about your Internet search and viewing habits. Lord knows a lot of companies are trying to do the same.
On the other hand, if you use Gmail and other Google services, it's fair to say that Google enjoys access to an extraordinary amount of data about you. Electronic Frontier says your YouTube habits "can reveal particularly sensitive information about you, including your location, interests, age, sexual orientation, religion, and health concerns."
As for your general search history, EFF warns:
Until now, your Google Web History (your Google searches and sites visited) was cordoned off from Google's other products. This protection was especially important because search data can reveal particularly sensitive information about you, including facts about your location, interests, age, sexual orientation, religion, health concerns, and more. If you want to keep Google from combining your Web History with the data they have gathered about you in their other products, such as YouTube or Google Plus, you may want to remove all items from your Web History and stop your Web History from being recorded in the future.
The letter from the attorneys general says:
Maybe you don't believe you have the choice to stay away - not if Google has you hooked into its ecosystem, as it certainly has me.
But if you're concerned about everything it knows, or just don't want to see all its inferences about you in ads or offers, at least you can start tomorrow with a cleaner slate.