Although some countries will transfer your Visa from your old passport to your new own, this is not necessary and will cost a fee. Do not attempt to cut your Visa out of your expired passport and put it in your new one - this could render the Visa invalid. Most countries (the United States, http://travel.state.gov/visa/questions/questions_1253.html, and the United Kingdom, http://www.ukvisas.gov.uk/en/faqs/, for example) will advise that you travel with both your expired passport containing the valid Visa, and your new passport, as long as both passports were issued by the same country.
Upon arrival at your destination, the Immigration Officer will check your Visa in your expired passport, as well as your new passport to allow entry. It is always best to check with the government of the country you are visiting to make sure you are following proper protocol. Project Visa, http://projectvisa.com/, provides Visa and Embassy information for all countries.