Google has several projects including Google Maps, Google Sites, Google Groups, and even Google Mars.
It's mission statement is:

Google's mission is to organize the world's information and make it universally accessible and useful.

This page will mostly explain how to use Google's search box to maximize the efficiency of returning the results you want.

Google uses its own high tech system. Most information about its system is top secret. It is perhaps this that makes Google the most effective search engine today. The purpose of PageRank is to show the most likely page containing your query first. This is key, since returning pages in any other way, such as alphabetical, or publish date, is practically useless. The formula, called the PageRank Algorithm, is very top secret.
But basically, Google uses a sort of point system. Depending on the page's popularity, and the number of links directed towards it. The highest scoring pages are shown first. This, surprisingly, works quite well.

Entering your Query
1) Remember to only enter KEYWORDS. Refrain from typing full questions.
Ex: If I were to search, trying to find out what PageRank is, I would type "PageRank" versus "What is PageRank?"

2) Order matters. Google reads your query from left to right, the first word being the most important and the last word being the least important.
Try it: Type in a multi-word search such as "Cupertino Middle". Then, search the same words in a different order, "Middle Cupertino". You will receive slightly different results, although usually, the top results are similar, it is the later results that alter.
Note: Sometimes, ordering your words may help.
For example: Consider - "City of Sunnyvale in California". Better may be - "Sunnyvale California City of in".
Although it doesn't grammatically make much sense, it may subtly return more desired results.

3) Caps don't matter.
Try it: Search "California" then "california" the results will be the same.

4) AND. Google assumes the word AND between every word you type.
Ex: Cats and dogs can be searched for by simply entering "cats dogs".  AND is assumed in between cats and dogs, and caps don't matter.

5) OR. Since Google assumes AND in between every word, to search for either this or that, add in the OR operator. OR must be in full caps, otherwise, it will be ignored (see 8).
Ex: Cats or dogs can be searched for by simply entering "cats OR dogs". OR is then used instead of AND and results for either cats or dogs will appear, however cat results may appear first since that is the first word.

6) -. To exclude a word from your searches, add in the "-" operator.
Ex: By searching bass, you will get results about music and fish. If you only want music, you could throw in the word music first ("music bass") or you can exclude fish (bass -fish).

7) Letter Wildcards. In most search engines you can use a * to indicate a wildcard. But not Google. Its automatic.
Ex: For most engines like Yahoo!, typing "book*" would return "books", "bookstore", "bookkeeper". In Google, simply typing "book" will automatically return the same results.

8) Word Wildcards. You can however use wildcards to stand for whole words.
Ex: If you are surching for MLK Jr.'s "I Have a Dream Speech", but you forgot some of the words, you could search "i have * * speech". You could use multiple wildcards, but too many will make your search worthless. "i * * *".

9) Excluded words. AKA "stop words". Google ignores these words in your search.
These include: and (it is already assumed, see 4), the, where, how, what, or (if not in full caps, see 5), etc., certain single digits, and certain single letters ("a").

10) Insisting on Stop Words. If you really want to search using a stop word, enter the + operator before it and it will be included in the results.
Ex: "+Where +is Sunnyvale, California".

11) Exact phrase. If you are searching for something like a name, or a quote, it may be wise to enclose it in " " marks to tell Google you are looking for that phrase and it must be in that order.
Ex: " "barnum and bailey the greatest show on earth" " versus "barnum +and bailey +the greatest show +on earth".

12) Words that don't appear together. If you are searching for a document, and forgot what's in between to words, you could search using this format....
Ex: " "dear joe, * sincerely dave" "

13) Specific filetypes. Use the filetype operator to search for specific types of files.
Ex: If you are searching for only PDF's, type "filetype:pdf airplanes"
Ex2: If you do not want PDF's, type "-filetype:pdf airplanes"

Here's a list of extensions to plug in to your searches:
Adobe Portable Document Format (pdf)
Adobe PostScript (ps)
Autodesk (dwf)
Google Earth (kml OR kmz)
Lotus 1-2-3 (wk1 OR wk2 OR wk3 OR wk4 OR wk5 OR wki OR wks OR wku)
Lotus WordPro (lwp)
MacWrite (mw)
Microsoft Excel (xls)
Microsoft PowerPoint (ppt)
Microsoft Word (doc)
Microsoft Works (wdb OR wks OR wps)
Microsoft Write (wri)
Rich Text Format (rtf)
shockwave flash (swf)
Text (ans OR txt)

14) Specific domain. Use the site operator.
Ex: To search only .edu websites: " space"

15) Specific site. Use the site operator again.
Ex: "site: Googling"

16) Searching only page titles. Use intitle for one word, and allintitle for multiple words.
Ex: "intitle:honda"
Ex2: "allintitle:honda ford toyota"

17) Searching only url's. Use inurl and allinurl in a similar manner.
Ex: "inurl:google"
Ex2: "allinurl:google sites thehynescmsbears"

18) Searching only body text. Use intext and allintext, again, in the same way.
Ex: "intext:googling"
Ex2: "allintext:googling pagerank overview"

19) Searching for a range of numbers. Use the ... operator.
Ex: Search for all numbers between 100 and 150. "100...150"

20) Searching for pages linking to a specific page. This will display all pages that have a link, linking to the page you enter.
Ex: link:

That's It!
Try it out below or on Google's homepage.