Since April 1 2015, you can apply online and complete your application at an Estonian embassy. All you need is your passport or photo ID.
When filling in the form online, you'll need your name and identifying information, a scanned passport photo, a scanned copy of government-issued identity documentation, a minimum of one paragraph describing your interest and motivations behind applying for e-residency, and payment of the state fee in the amount of €100 via Visa or Mastercard.
Then comes the background check, when your documents are sent to Estonia. This helps the police and border guard board with risk management and the background check. You may also be asked if you have any previous links with the country.
After about four weeks or so, you'll receive an email to let you know whether your application was successful. If it's a yes, you can go and pick up your identity card from one of 38 embassies.