Fill-in forms.
The simplest type of interactivity. In their bare-bones form all they require is HTML. More commonly you will need a CGI script to process the form data. This type of interactivity is excellent for surveys, polls, requests for comments, and the like.

For example: An NYU form for requesting an application.

Threaded discussion groups, aka, BBS.
These are a frequent addition to web sites, with mixed results. A BBS is a fragile thing. People only seem to post when there is already activity in a discussion group, so you will need to carefully seed and maintain a BBS if you include one in a site. Somehow threaded discussions seem to happen more successfully in newsgroups or on mailing lists.

For example: Slashdot

If you do want to include a BBS in your site you will likely need to use CGI scripts. There are some available in the public domain which you can download from sites like Matt's Script Archive for free. If you want the convenience of easy administration tools, or layout flexibility you may want to do a search for a commercial package with the features you need.

Guestbooks.
Similar to a BBS, but generally these are used to get people to respond to one thing, such as an entire site, or an article. There's a short form, people type in their response, and it gets added to a list of other people who posted before.

For example: The Fray uses guestbooks after each story.

Chat rooms.
Chat has long been exceedingly popular with young people, and people who have extra time on their hands. Lately chat has emerged from IRC (Internet Relay Chat) and AOL and is being used for tech support by some major companies. There are a variety of Java or CGI solutions which can incorporate chat into a web site. Or, there are standalone applications such as The Palace which you can use separately but in association with a web site.

For example: Yahoo Chat.

eCommerce - Online catalog and/or ordering.
The catalog part is pretty straightforward. In simplest form, you could just create a static table to illustrate the products. A more sophisticated solution would work with an existing product database, to which you would link from web pages. To create a dynamic link to a database you will combine fill-in forms for specifying specific items, which pass data to a CGI (or analagous) script called "middleware" which re-formats the HTML from the form into a query which the database can understand (SQL for example.) When the database yields a result it is passed back to the middleware which re-formats the result as HTML and sends it on to the browser.

There are a variety of middleware packages tailored to different database packages. The easiest to use is probably Access, with SQLServer, Oracle, or Informix databases representing the industrial power databases. mySQL, mSQL, and PostGres are freeware databases which are open source alternatives to the commercial packages.

For example: Dell Computer Store.

Interactive games and comics.
Often used as a fun add-on to generate interest in a site. Games will generally require programming in Shockwave or Java, requiring that users have up-to-date browsers.

For example: various things on Macromedia Shockwave. (check out Stain Boy)

Streaming Media.
The Internet is only beginning to make a good presentation medium for audio and video, but streaming formats are finally starting to get good. Especially audio-only which sounds better over slow connections There are a variety of technologies such as Real, Shockwave, ActiveMovie, Vivo, etc. for distributing audio and video content. The length or clips that you want to present will help you decide whic technology to use. Also, they have very different costs.

For example: KCRW broadcasts a bunch of their programming now on the web.