What you’re looking for is a System that checks when you’ve changed the selected element within your Jlist.
You can make your Jlist implement a Listselectionlistener, and overwrite the method valueChanged().
Example:
import java.awt.BorderLayout;
import java.awt.EventQueue;
import javax.swing.JFrame;
import javax.swing.JPanel;
import javax.swing.border.EmptyBorder;
import javax.swing.event.ListSelectionEvent;
import javax.swing.event.ListSelectionListener;
import javax.swing.JList;
import javax.swing.JLabel;
public class Lista extends JFrame {
    private static final long serialVersionUID = 1L;
    private JPanel contentPane;
    private JLabel lblNewLabel;
    private JList<String> list;
    public static void main(String[] args) {
        EventQueue.invokeLater(new Runnable() {
            public void run() {
                try {
                    Lista frame = new Lista();
                    frame.setVisible(true);
                } catch (Exception e) {
                    e.printStackTrace();
                }
            }
        });
    }
    public Lista() {
        setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);
        setBounds(100, 100, 450, 300);
        contentPane = new JPanel();
        contentPane.setBorder(new EmptyBorder(5, 5, 5, 5));
        contentPane.setLayout(new BorderLayout(0, 0));
        setContentPane(contentPane);
        lblNewLabel = new JLabel("New label");
        contentPane.add(lblNewLabel, BorderLayout.SOUTH);
        list = new JList<String>();
        list.setListData(new String[]{"Cachorro", "Elefante", "Gato"});
        list.addListSelectionListener(new ListSelectionListener() {
            @Override
            public void valueChanged(ListSelectionEvent e) {
                System.out.println(e);
                if (list.getSelectedValue() != null) {
                    lblNewLabel.setText(list.getSelectedValue().toString());
                }               
            }
        });
        contentPane.add(list, BorderLayout.CENTER);
    }
}
Every time you change the Jlist element it updates the Jlabel below it.
Prior:
