The European Library offers access to the resources of 47 European national libraries, as well as an overview on accessible digital collections. See the rather long and boring tutorial video on searching the European Library catalog below - or give it a try it on your own.
See the link list below for National Libraries which offer publicly accessible digital library projects.
If a country is not listed here, it may not be available online or available for searching in English. See the National Libraries of the World list to look on the websites of the national libraries.