Am planning migration of Intranet from MOSS 2007 Standard to SharePoint 2010 Standard. Do I actually need the Secure Store to access external database? For example, when pulling data for a Web Part... am I required to use the Secure Store?
It depends what credentials you need to access the external database!
If you need to access it with the credentials of the current user then you must either use Kerberos authentication or use the Secure Store to store user credentials for accessing the external database. Kerberos delegates the identity token all the way from the client to the external database and you will experience true single sign-on. With the Secure Store users will have to enter their credentials the first time they access the external database. I can recommend that you also read the TechNet article Plan the Secure Store Service (SharePoint Server 2010)
You will not need Kerberos or the Secure Store if the user context does not matter. In that case you can simply access the external database with the IIS application pool account, i.e. use SPSecurity.RunWithElevatedPrivileges in your Web Part.