Field type "date" is not in the list of available field types I'm trying to create a new content type with a field that would contain a date. However, the drop-down list for field type doesn't have "date". What's wrong?

p.s. drupal 7.17
You need to install the date module:

http://drupal.org/project/date

Then enable the date module and that will give you the option to add a date field to your content type. Depending on your case you might want to check also the popup calendar module that is part of the date module to ease the date input.