If e-mail provider used POP3 server, mail was downloaded to your computer and will have to be exported to your new laptop. A provider with IMAP server retains mail on server so just setup account with same e-mail address and password in your e-mail client. Your e-mail is copied to your computer when you open this account.
Have you tried using Windows Easy Transfer Tool? It should make the process relatively simple, although you have to find your Outlook files first. I'm assuming you are talking about Outlook since you tagged this as a Windows question.