Database Design for Tagging
I don't see a problem with a straightforward solution: Table for items, table for tags, crosstable for "tagging"
Indices on cross table should be enough optimisation. Selecting appropriate items would be
SELECT * FROM items WHERE id IN
(SELECT DISTINCT item_id FROM item_tag WHERE
tag_id = tag1 OR tag_id = tag2 OR ...)
AND tagging would be
SELECT * FROM items WHERE
EXISTS (SELECT 1 FROM item_tag WHERE id = item_id AND tag_id = tag1)
AND EXISTS (SELECT 1 FROM item_tag WHERE id = item_id AND tag_id = tag2)
AND ...
which is admittedly, not so efficient for large number of comparing tags. If you are to maintain tag count in memory, you could make query to start with tags that are not often, so AND sequence would be evaluated quicker. Depending on expected number of tags to be matched against and expectancy of matching any single of them this could be OK solution, if you are to match 20 tags, and expect that some random item will match 15 of them, then this would still be heavy on a database.
About ANDing: It sounds like you are looking for the "relational division" operation. This article covers relational division in concise and yet comprehendible way.
About performance: A bitmap-based approach intuitively sounds like it will suit the situation well. However, I'm not convinced it's a good idea to implement bitmap indexing "manually", like digiguru suggests: It sounds like a complicated situation whenever new tags are added(?) But some DBMSes (including Oracle) offer bitmap indexes which may somehow be of use, because a built-in indexing system does away with the potential complexity of index maintenance; additionally, a DBMS offering bitmap indexes should be able to consider them in a proper when when performing the query plan.
Here's a good article on tagging Database schemas:
http://howto.philippkeller.com/2005/04/24/Tags-Database-schemas/
along with performance tests:
http://howto.philippkeller.com/2005/06/19/Tagsystems-performance-tests/
Note that the conclusions there are very specific to MySQL, which (at least in 2005 at the time that was written) had very poor full text indexing characteristics.
I just wanted to highlight that the article that @Jeff Atwood links to (http://howto.philippkeller.com/2005/04/24/Tags-Database-schemas/) is very thorough (It discusses the merits of 3 different schema approaches) and has a good solution for the AND queries that will usually perform better than what has been mentioned here so far (i.e. it doesn't use a correlated subquery for each term). Also lots of good stuff in the comments.
ps - The approach that everyone is talking about here is referred to as the "Toxi" solution in the article.