After you have configured a server in eclipse, you can further configure it and add/remove/tweak any modules (aka contexts/webapps) that you want it to host. You should have a Servers window that list the server (if not: Window > Show View > Servers). Double click on on the server to open the config page. There are 2 tabs, but they are in the bottom left corner. Open Modules, any module you added will be listed. If you want a module to be the default one (so if you type localhost it will server index.jsp, for example) make sure that path says "/". The default value will be what is in your webapps Properties > Web Project Settings, and it usually be the project name. If you set it there, it will not let you leave it blank, so use "/".
Make sure you save the config changes for your Server. You may also need to "publish" it to make the changes propagate out to the actual server.
A last thing to check is find the server install, open conf/server.xml and look for your contexts. You can change the path there as well. Note that path="" equates to "/", tomcat will make that the ROOT app. Also note that Eclipse will create an empty webapp named ROOT when you first publish it (this will also be in server.xml). It is safe to delete it. Even though your deployed app will have the project name as the webapp name, as long as the path is "", it will treat it as the default ROOT.