DataTemplates

As styles aren't stored in Resources, neither are DataTemplates. Instead, DataTemplates are placed in a DataTemplates collection on each control (and on Application):

<UserControl xmlns:viewmodels="clr-namespace:MyApp.ViewModels;assembly=MyApp">
    <UserControl.DataTemplates>
        <DataTemplate DataType="viewmodels:FooViewModel">
            <Border Background="Red" CornerRadius="8">
                <TextBox Text="{Binding Name}"/>
            </Border>
        </DataTemplate>
    </UserControl.DataTemplates>
    <!-- Assuming that DataContext.Foo is an object of type
         MyApp.ViewModels.FooViewModel then a red border with a corner
         radius of 8 containing a TextBox will be displayed here -->
    <ContentControl Content="{Binding Foo}"/>
<UserControl>

Data templates in Avalonia can also target interfaces and derived classes (which cannot be done in WPF) and so the order of DataTemplates can be important: DataTemplates within the same collection are evaluated in declaration order so you need to place them from most-specific to least-specific as you would in code.

DataTemplateSelector

In WPF you can create a DataTemplateSelector to select or create a DataTemplate based on the provided data. In Avalonia you cannot do this, but you can implement IDataTemplate which can be seen as a good replacement for the DataTemplateSelector. Please find a sample here.