So your organization has a Twitter account - but it's over there on Twitter.com. Just like your organization's blog, you need to integrate your Twitter feed with your website. If your organization uses a content management system (CMS), it is likely that an RSS/Newsfeed display module is included.
If you are not using a CMS or yours does not have an RSS module included, you can still integrate your Twitter (of other newsfeeds) into your website. There are several ways to do that, and it gets easier all of the time. The easiest way is to use a widget. Here are examples of several that we have come across. They are easy to use - you can set them up in a matter of minutes. They all have different looks and feels - usually customizable - so try a few and see which ones work for you.