Given a keyword, how do I obtain information on how many times it's
been googled (or looked for at any other engine) per day.

Hello clickclick-ga,

the nearest you will find is Overture's search term suggestion tool
http://inventory.overture.com/d/searchinventory/suggestion/
which is used by advertisers to determine the most popular keywords.
This also acts as an indicator for keywords in general, as the
Overture search content and results is served across many other major
portals:
- Yahoo! 
-	MSN 
-	Lycos 
-	AltaVista 
-	InfoSpace 
-	United Online 
-	iWon
(from Overture's major affiliate partners list) 
http://www.overture.com/d/USm/about/news/glance.jhtml

The closest Google come is with the Traffic Estimator which serves the
purposes of advertisers:
"Traffic Estimator - After you select your keywords, the AdWords
Traffic Estimator automatically provides you with estimates for the
number of clicks per day, average cost-per-click, cost per day, and
average ad position for each of your keywords." excerpt from
Adwords.google page
https://adwords.google.com/select/terms.html

The Overture tool is currently the most comprehensive benchmark for
keyword popularity, with the widest scope across it's partner sites.

I hope that helps, if you need any clarification of the answer just
ask.
Kind regards
lot-ga

Hi lot-ga
Thanks a lot for your answer. The thing with adwords' traffic
estimator I knew already, but I didn't know about Overture. Well done.

Word Tracker is also a very good service at providing keyword access
information. You can learn more about their service at
http://www.wordtracker.com