I would like to display an alert on my web site.
I could use a library like jQuery, but currently, the site does not use one, and I don’t think it is a good idea to add a library just for the sole purpose of an alert box.
Is there any way of styling the alert box, or is there a pure js alternative to something like jQuery modal dialog?
No, it is browser chrome and you can’t touch it.
jQuery doesn’t have a modal dialog. jQuery UI does.
There is no pure JS alternative — you are dealing in presentation, so you need CSS.
jQuery UI is pure JS + CSS, and anything you can do with it, you can do by writing the JS and CSS yourself. Look at their code if you want an example.