Unless you delete your account you cant delete the inbox, but you can delete the messages that are inside the inbox.
You can either:
1.) Delete them- moves the emails to your trash where you can delete forever from there
2.) Archive them- this takes the emails out of your inbox and just keeps them in the "All mail" folder.
3.) Move them- you can move them into a folder that you created.
If you want to just keep one or two you will have to individually check the boxes you dont want and leave out the ones that you do. Or if you want to delete all of your messages, you can select "all" from the dropdown list that comes from the box at the top bar of your inbox (it is at the top right).
Once the emails are selected (either all of them, or just some) you can choose what you want to do with them ( Delte,archive,move).
If you want to delete your account,
Go to settings at the top right of your gmail page > click google account settings > click edit (to the right of my products in the pop up window) > under delete account hit "Close account and delete all services and info associated with it"