An input dialog asks for one line of text. Like the other dialogs it is drawn inside the MetroWindow rather than in a window of its own.
ShowInputAsync returns the text that was entered, or null if the user backed out:
private async void OnButtonClick(object sender, RoutedEventArgs e)
{
var name = await this.ShowInputAsync("What is your name?", "This will appear on your profile.");
if (name is null)
{
return; // cancelled
}
// use name
}

From a view model, use the dialog coordinator instead of reaching for the window.
The null result
Everything that is not "the user pressed the affirmative button" gives you null: the negative button, Esc, Alt+F4, and cancelling the CancellationToken.
null and an empty string are different answers. An empty string means the user confirmed an empty box; null means there was no answer at all. Check for null rather than using string.IsNullOrEmpty if the two should behave differently.
Prefilling the box
DefaultText puts a value in the box before the dialog opens. It arrives selected, so typing replaces it and confirming keeps it:
var settings = new MetroDialogSettings
{
DefaultText = "Ada Lovelace",
AffirmativeButtonText = "Save",
NegativeButtonText = "Skip"
};
var name = await this.ShowInputAsync("What is your name?", "This will appear on your profile.", settings);
Colour scheme
ColorScheme works as it does for the other dialogs: Theme follows the current theme, Accented fills the dialog with the accent colour, Inverted uses the inverse of the theme.

Which settings actually apply
MetroDialogSettings is shared by all the dialog types, and an input dialog reads only part of it. The rest is accepted and ignored, which is worth knowing before you spend time on a setting that cannot take effect here.
| Setting | Effect on an input dialog |
|---|---|
DefaultText |
prefills the box |
AffirmativeButtonText |
label of the confirm button, OK by default |
NegativeButtonText |
label of the cancel button, Cancel by default |
ColorScheme |
as above |
DialogTitleFontSize, DialogMessageFontSize, DialogButtonFontSize |
as for the other dialogs |
AnimateShow, AnimateHide |
as for the other dialogs |
OwnerCanCloseWithDialog |
whether the window can be closed while the dialog is up |
CancellationToken |
closes the dialog; the call returns null |
CustomResourceDictionary |
resources for the dialog |
FirstAuxiliaryButtonText, SecondAuxiliaryButtonText |
ignored — an input dialog has exactly two buttons |
DefaultButtonFocus |
ignored — focus starts in the text box |
DialogResultOnCancel |
ignored — cancelling always gives null |
MaximumBodyHeight |
ignored — only the message dialog uses it |
Keyboard
Focus starts in the text box, so the user can type straight away. Enter confirms and returns the text. Esc cancels and returns null.
Outside a MetroWindow
Where there is no MetroWindow to draw into, ShowModalInputExternal opens the dialog in a window of its own and blocks until it is answered:
string name = this.ShowModalInputExternal("What is your name?", "This will appear on your profile.");
Being synchronous, it is not the one to use from an async path.