Google's indexing behavior is optimized for quickly finding new pages, not so much for noticing the disappearance of old pages. Generally, this makes sense — when you search Google, you want to find something, so not finding a new page that has just appeared is normally a much bigger failure than occasionally being sent to a page that used to be there, but isn't any more. Besides, most popular web pages don't usually disappear, and if they do, that's often just a temporary issue caused by a server hiccup.
In particular, Google will normally retain removed pages in its index until:
it happens to re-crawl the page and notices that it returns a
404 response, and
this happens sufficiently many times in a row that Google's algorithm decides that the page probably isn't coming back.
Thus, it's not uncommon for removed pages to linger for weeks or even months in Google's index, especially if the site they were on is relatively low-ranking and therefore infrequently re-crawled by Google.
In those rare cases where you do want to have a page quickly removed from Google's index, there are a couple of things you can do:
You can ask the owner of the site to submit a removal request for the page via Google's Webmaster Tools.
If the page has already been removed from the site, you can use the Google public URL removal tool yourself to tell Google to recrawl the page and remove it from their index.
Both of the above methods should result in the page being removed from Google's index within a day or so. Note that, if the page has been indexed under multiple URLs, you'll need to file removal requests for all of them.
For completeness, as John Conde suggests in his answer, you can ask the webmaster of the site the page was on to configure their web server to return an
410 Gone response for the page instead of the usual
404. This will not make Google re-crawl the page any faster, but when it does, the
410 reponse does tell Google that the page probably isn't coming back.
Also, don't forget that Google isn't the only search engine out there. To get the content completely off the web, you'll also need to file removal requests with Bing, Yandex and any other search engines that may have indexed it. (In some cases, you may be out of luck; for example, the only URL removal instructions I could find for Baidu involve sending e-mail to a helpdesk address.) And don't forget archives like the Wayback Machine, either; while such archives are not searchable, someone who knew where the page used to be might still be able to find an archived copy. It's up to you to decide whether or not you consider that an issue.