You should use "Interactivity" assembly and SelectionChanged event.
<DataGrid AutoGenerateColumns="False" ItemsSource="{Binding People}">
    <DataGrid.Columns>
        <DataGridTextColumn Header="ID" Binding="{Binding ID}" />
        <DataGridTextColumn Header="Name" Binding="{Binding Name}" />
    </DataGrid.Columns>
    <i:Interaction.Triggers>
        <i:EventTrigger EventName="SelectionChanged">
            <i:InvokeCommandAction Command="{Binding MyCommand}" />
        </i:EventTrigger>
    </i:Interaction.Triggers>
</DataGrid>
Where "i" is namespace:
xmlns:i="clr-namespace:System.Windows.Interactivity;assembly=System.Windows.Interactivity"
Also you can write binding to SelectedItem property of the DataGrid and in the set accessor you can invoke your command, but the first solution that i presented you above is better.
If you want to invoke command from main view model and pass SelectedItem from DataGrid you can use CommadParameter:
<i:Interaction.Triggers>
    <i:EventTrigger EventName="SelectionChanged">
        <i:InvokeCommandAction Command="{Binding MyCommand}" 
        CommandParameter="{Binding Path=SelectedItem, RelativeSource={RelativeSource Mode=FindAncestor, AncestorType=DataGrid}}" />
    </i:EventTrigger>
</i:Interaction.Triggers>
When items has own command you can use following code:
<i:Interaction.Triggers>
    <i:EventTrigger EventName="SelectionChanged">
        <i:InvokeCommandAction Command="{Binding Path=SelectedItem.MyCommand, RelativeSource={RelativeSource Mode=FindAncestor, AncestorType=DataGrid}}" />
    </i:EventTrigger>
</i:Interaction.Triggers>
Or if elements has own view model that is assigned to it's DataContext, you can use following code:
 <i:Interaction.Triggers>
    <i:EventTrigger EventName="SelectionChanged">
        <i:InvokeCommandAction Command="{Binding Path=SelectedItem.DataContext.MyCommand, RelativeSource={RelativeSource Mode=FindAncestor, AncestorType=DataGrid}}" />
    </i:EventTrigger>
</i:Interaction.Triggers>