Each time the app is launched, contacts on the server are synced with the phone's address book. Contacts with Telegram accounts are shown first in list, followed by the rest of the contacts on the device.
Tapping a Telegram contact opens a chat with the user. Tapping an unregistered contact opens the invitation dialog — the standard SMS sending screeen. The input field should contain a standard text with a link, inviting the user to install the app.
This screen can be accessed by tapping on: