You can get the selected text of a TextBox through the property SelectedText.
You can put something on the clipboard using the static method:
Clipboard.SetText(textBox1.SelectedText);
And get text from the clipboard (also the one copied into the clipboard in the browser) with:
if (Clipboard.ContainsText()) {
textBox1.Text = Clipboard.GetText();
}
See: Clipboard Class.
If you only want to do that on the current form, you can get the current control with the form property ActiveControl. But will have to use different methods for different types of controls in order to get the selected text. Controls deriving from TextBoxBase have a SelectedText property. All controls have a Text property. All controls deriving from ListControl (like ComboBox) have a SelectedValue property. ComboBox has a SelectedItem property.
As you can see, you will have to make experiments and find the strategy which works best.
Also have a look into the Object Browser of Visual Studio, especially into the namespace System.Windows.Forms.