Embedding a widget into a wiki or web page is a relatively simple process.
The concept is to ensure that you always have fresh content on your page. You might do this by echoing your own Twitter messages, your favourites, your lists of users, that of your organization, or a hashtag for an event that you're monitoring or anything you can search on Twitter.
Twitter itself provides a number of widgets available for embedding here.
In this case, I'm just going to create a widget for my Twitter ID.
Click the "Create widget" button and Twitter will generate the code for you. Copy that and insert into your web resource. You may have to grant extra permissions to run the widget.
Save it and you're live! Note that the image above was a picture I took of the screen as I created the widget. The live widget appears below.