The most important facts have already been mentioned in the answer of Patric Hartmann. However, I'd like to add some further information.
Töten is sort of generic term and is defined as to take someone's life (in German: jemandem das Leben nehmen). This word on its own does not contain any further hints on how the killing has been executed. And töten also includes death caused by, for example, natural disaster.
Großes Tokio-Erdbeben könnte 23.000 Menschen töten. Quelle
This is not possible with ermorden and umbringen, because both words include the connotation of doing the act of murdering intentionally.
The first word, ermorden, does also–in my humble opinion–slightly carry the idea of doing that in an cruel or malicious way (regardless that murdering someone is always a terrible criminal act). I mean, ermorden often comes with brutal or grausam and is also a common word when talking about Taliban or such, murdered presidents and rape.
Diese US-Präsidenten wurden ermordet. Quelle
The latter one, umbringen, is also quite generic. This word is most commonly used for suicide, which is rarely described with töten or ermorden. And according to corpora, this word is also the only one commonly used for killing each other. Other than that, it's quite neutral in its meaning and does not carry any further connotations.
Ihre Tochter wollte sich umbringen! Quelle
Apart, umbringen is very common in figurative expressions like "Du bringst mich noch um.", which just means that someone drives you crazy.
I used the corpus Wortschatz-Portal to ensure that my claims about collocations are based on facts. ;)