I can think of two options using Gmail:
1. Select the email you want to remove from your inbox then click on the little down arrow on the Move to button (looks like a folder). In there you can select an existing folder or click 'Create new' to make a new one and your email will be moved there. It won't be removed from your account but will no longer appear in your inbox. If there any attachments, they will be moved too.
You can retrieve the email/attachements in future by opening the folder from the list on the left.
Alternatively:
2. Open the email you want to remove from your inbox and then click the little printer icon that appears to the right hand side of the email title. This will open a new window. Click the Change button shown near Destination and then select Save as pdf. The window will close and you'll be back at the Print window. Click on Save and then select the folder you want to save the email to. You can change the file name to something more meaningful if you want to and then click save. This will convert the email to a pdf file (which makes it difficult to amend). You can then delete the email from your inbox because it's safe in it's new location (as long as you back up regularly, like any other files). If there are any attachments to your email, it's a good idea to save them separately. It saves the whole email including the data/time it was sent, the sender and the recipients.
Hope that helps.
CathieM