Email addresses are not tied to a computer. they are tied to your email provider. if you use webmail (ie hotmail, gmail, etc) then you can simply access your email from any computer.
If you are retrieving your mail using an email client like Outlook or Mozilla Thunderbird, then the email client will need to be installed on the new computer. You will also need to backup your mail from the old computer and move it to the new computer which will probably be out of your ISP's scope of support but you can find instructions on how to do that on google based on which email client you are using.