If your emails disappear from your Inbox after replying to them, you probably have Gmail's Send and Archive option enabled.  Having Send and Archive enabled gives you two send buttons when you reply to an email.

Hide the blue Send + Archive button

1. Sign in to Gmail.
2. Click Settings.
3. Look for the Send And Archive setting and select the radio button next to the Hide "Send & Archive" button in reply.
4. Click Save Changes at the bottom of the page.

For more information on archiving messages in Gmail, see Google's support article Archive or mute Gmail messages.