Console
See the Wikipedia page on ANSI escapes for the full collection of sequences, including the colors.
But for one simple example (Printing in red) in Java (as you tagged this as Java) do:
System.out.println("\u001B31;1mhello world!");
The 3 indicates change color, the first 1 indicates red (green would be 2) and the second 1 indicates do it in "bright" mode.
GUI
However, if you want to print to a GUI the easiest way is to use html:
JEditorPane pane = new new JEditorPane();
pane.setText("hello world!");
For more details on this sort of thing, see the Swing Tutorial. It is also possible by using styles in a JTextPane. Here is a helpful example of code to do this easily with a JTextPane (added from helpful comment).
JTextArea is a single coloured Text component, as described here. It can only display in one color. You can set the color for the whole JTextArea like this:
JTextArea area = new JTextArea("hello world");
area.setForeground(Color.red)
