A visa granting leave to stay in the UK does not allow you to travel to other countries in the EU.
Each country in the EU is an independent nation state with its own independent immigration rules. Check the website of the relevant country's consulate to find out what you need to do to travel to that country.
However, most countries within the EU, as well as Norway and Switzerland, are members of the 'Schengen Area'. One Schengen visa will allow you entry to all of these countries.
For more information about obtaining a Schengen visa, visit the website of the embassy of the country or countries you wish to travel to.
If you plan to travel to more than one of the countries listed below and you are travelling for less than 3 months as a tourist (not working), the Schengen Agreement allows you to use one visa to enter all of the signatory states. For more information, see http://www.beds.ac.uk/international/international-student-support/visas-and-immigration/traveling-during-your-studies