Question: Is it possible to use SharePoint object model to enumerate all sites in an application to determine what the base site definition is? (and also to know if the site was provisioned from a template?)
thx

Answer: 
You can iterate your site collections and examine every SPWeb to discover WebTemplate, WebTemplateId and Configuration.

These values match the template and configuration used to create the site (aka web in API). You can then compare that against the site definitions found in SharePoint root (14 hive if u wish) in the respective ONET.XML files and webtemp\*.xml files.

<http://msdn.microsoft.com/en-us/library/microsoft.sharepoint.spweb_members.aspx>

In SP2007 there was **no** way to discover if your site was provisioned by a site definition template or a template file (STP). Since STP files are now retired (for site templates anyway) in favor of WSP files, I would assume that its still impossible to see this.
To add to Anders answer, you can get the name through the API, too in 2010 and 2013. 

Use [SPWeb.GetAvailableWebTemplates](http://msdn.microsoft.com/en-us/library/microsoft.sharepoint.spweb.getavailablewebtemplates%28v=office.12%29.aspx). Another SharePoint.stackexchange.com thread has [a great example of using the API](https://sharepoint.stackexchange.com/questions/5667/which-site-template-is-a-site-using) through PowerShell.