This is a question to those who have AdSense and use the Google search on there site:
why do you have the google search box leave your website when you click submit (aka it goes to http://www.google.com/custom?q=....)?

Google has some APIs which allows you to query there servers and display the results on your website so that you never have to let the user leave your website. That and you can display the search page however you want. Are there some benefits to usign the custom one versus usign there API's (and completly customizing it)? And yes u can restrict the searches to just your website(s) or the entire web, it's really all up to you!

thats pretty cool, are there written documents on how you can do that? or is it in the web dev. script (html, javascript etc.)

they give some examples in there api (including some in java and C++)
http://www.google.com/apis/

you could follow those (if not I could give a copy of the source code I used)

Using Google's (non-API) version is incredibly easy to implement and you get money when searchers click on sponsored links. When using the API version, none of the sponsored links show up (and until recently, google prohibited displaying AdSense on search result pages.) Of course you need programming experience to use the API version. But aside from that, one of the main reasons I don't use it here is because Google restricts you to a maximum number of searches per day, which isn't always practical with largescale sites.

I have a version of Google Search using the API on my site. I tried to integrate it to give the same features and feel of the actual Google Search page.

http://www.vebro.net/websearch.asp

I remember reading that Google's API restricted you to a maximum number of searches per day. Is this true? How limiting is Google's API, exactly?

My version combines ASP and the Google API so using some tricky coding technique's I was able to throw together a nice usable search page.

As for the limit number of Google's API, that is true and is limited to 1,000 queries per day. (http://www.google.com/apis/api_faq.html#gen7)

you are limited to 1000 queries a day, and only 10 results per query...
however you can specify an offset like VSBrown did to display more results... another thing you could do if you wanted to get 50 or 100 results in 1 page is just put the code in a loop and get 10 results each time so you could put as many as you want on a page