Learn German: Sie wollte ihn umbringen.

All three mean to kill, but they have different nuances. töten is the general, neutral term for "to kill" (including by accident or killing animals). ermorden specifically means "to murder" (premeditated and malicious). umbringen is a very common, slightly more colloquial word for intentionally killing a person, similar to "to kill" or "to do someone in" in everyday spoken English.

Why is it umbringen here and not umzubringen, since the English translation says "wanted to kill"?

Because wollen is a modal verb. In German, modal verbs (like wollen, können, müssen) always take a bare infinitive at the end of the sentence, without the word zu (to). You only use zu with non-modal verbs, for example: Sie versuchte, ihn umzubringen (She tried to kill him).

umbringen is a separable verb. Why isn't the prefix um- separated here?

Separable verbs only split when they are the conjugated, active verb in the sentence (for example, Sie bringt ihn um). In this sentence, the modal verb wollte is the conjugated verb in position two. This pushes umbringen to the very end of the clause in its full, unseparated infinitive form.