Searching for Phrases
Google searches use words to generate results. For example if you enter high school into the search bar, you will receive results including both high and school. By applying quotation marks to the search, entering "high school" into the bar, Google will know to only search for results specifically including the phrase, not every appearance of the words "high" and "school."
Limiting Results
Typing a hyphen or minus sign in front of a word or phrase will exclude the words from the search results. For example "Apple -iPhone" searches for websites mentioning "Apple" but not "iPhone."
Looking for the Unknown
By typing in the phrase "better than _keyword_" or "reminds me of _keyword_" you can find ideas that you can't even name.
Find Related Terms
Adding a tilde (~) to a search term will find related terms. For example, Googling "~health" will give results that include nutrition, food and fitness.
Search Within a Website
Suppose you read an interesting article in the New York Times, but you can't remember the name or author of the article. Type "site:" followed by the URL of the website you'd like to search. Then add your search terms.
For more tips n' tricks...
Huffington Post
LifeHacker
The Official Google Guide