You have these possibilities (I'm not sure if the list is complete):
Negating the object:
Er spricht kein Englisch.
This means: English is not a member of the set of languages that he is able to speak.
Negating the verb:
Er spricht nicht Englisch.
This means: English is a language that he does not speak. (In my mind this is exactly the same meaning as in the first version.)
Negating the ability to do something:
Er kann nicht Englisch sprechen.
This means: He is not able to speak English. (This is different from the versions before. The first two versions contain the possibility that he is able to do it, but doesn't want to, or doesn't use it for other reasons. For example, there are some Jews who know how to speak German, but refuse to speak the »Tätersprache« (Language of the Nazis)
Since (at least in German) a language can also be interpreted as an activity that needs some skills, like skiing, painting, driving, you can also negate it like an activity:
Er kann nicht Englisch.
This means: He doesn't have the skills to speak English.