When you sign up for a free email account with Google's Gmail service, the address you select is permanent. Unfortunately, you cannot change this email address once the account is created. One reason Google prevents you from changing your email address is to stop emails bouncing back from people who have emailed you in the past and may email you again. You do have a few options though.
You are able to change the sender name that is attached to your Gmail email address. Although this change doesn't affect the email address, as that remains the same, you can change the sender name, which is seen by the recipient with the message. To change the sender name, go to "Mail Settings" section of your Gmail account, click the "Accounts and Import" tab and "Edit Info" link for the "Send Mail As" option. On the dialog box, change the name and click "Save Changes" button.
Under the "Send Mail As" setting, you can designate a reply-to address, which is different than your primary Gmail address. When the person receives your message, it'll show two email addresses. It'll show your primary address as the sender and show a reply-to address. If the person clicks the "Reply" button in his email application, then it'll compose a reply message that goes to the other email address.
Send From Different Address
Google lets you import other email accounts into Gmail. You can read messages from these other email accounts while using the familiar Gmail interface that you like. When you reply to one of these imported messages, it'll send from a different email address. You can access the "Accounts and Import" tab from the "Mail Settings" section to set up and import different email addresses.
Another option that you have is to forward your Gmail account to a different email address. This can be a second Gmail address or one at an entirely different service provider. Once you have forwarded your emails to the second email account, you can reply from that address, which shows your secondary email address with the message.
New Sign Up
One of the easier things you can do is to sign up for a second Gmail account. Google provides accounts free, so it won't cost you money out of your pocket. Also, Google doesn't limit how many Gmail addresses you may have. To sign up, log out of your first Gmail address, then go back to gmail.com and click the "Create an Account" button to get started.