Here's what you can do without doing any programming...
When looking at the map in ArcGIS.com, click Share on the menu bar. This allows you to do two things that may work. One, you can choose to embed the map in your own web page. However, there is no ability to turn layers on or off when embedded in this fashion. Two, you can make a web application that you deploy to your own web site. There are several templates to choose from, some that let you turn on/off layers.
Here's what you can do with some programming.
You can modify the template you downloaded in option 2 above, and edit the application (all files are included) to do exactly what you want. The templates are build (primarily) with the ArcGIS API for JavaScript.
We also have ArcGIS API for Flex and the ArcGIS API for Silverlight. All of the APIs (I believe) can read in a map you create in ArcGIS.com. You then can write your application from scratch, but leverage the map authoring abilities in ArcGIS.com
Hope this helps,
Mike
