There are many tools available on the internet to help you lookup an email address. Some tools (like this site) cost nothing; others have one time or monthly fees. If you want to lookup an address you need to know where to start.
What you must first realize about trying to find someone’s email address online is that if that sometimes a person does not want to be found, in this instance it may be more difficult or even impossible to find them. On the other hand, if the person you are looking for has maintained the same email address for some time and used that address to enter a contest, register a domain name or things online, there is a better chance this person's email can be found. The more time anyone spends using the email online, the more traces they may leave, and the easier it will be for you to find them.
With that said let’s examine a few of the ways other than using an email finding service one can use the internet to find an email address. You can always start by performing a search on your favorite search engines. When doing this be very specific with the details you search, use the full name of the person you are looking for and you may want to wrap quotes ("first last") around it. If you know additional information such as age, school, address, etc you may try adding that to your search query as well.
Often the more you know about someone the better. If you know the company or business this person works for, you may be able to do a search for the company name, see if they have a website and a public employee database. Many times these are made available for customers to write inquiries, questions, or comments. If they don't list the name or email address of the person you are looking for you can always try calling the business and asking for the person too, though I know it is not what you may have had in mind.
Finally, you may want to try the social networking sites to see if your person has a public profile. Sites like Myspace.com and Facebook.com may not publicly publish their members email addresses but they often provide a way to contact the person through the site.
Looking up an email address online is not always easy but with a little work it can be done.