I called him - usually audibly, either via telephone or just via voice
"I called him on my new cell phone" "I called him over to help us with the project"
I called on him - 2 different meanings
1. to visit (usually in person). "While in town, i called on my old friend Alex' (this usage is sort of quaint and is not used a lot)
2. to depend/rely on: "I called on Jack to help me finish the project" "To finish the project, we had to call on Jack's skills"
this isn't the full/proper explanation, so if someone wants to expand on it, feel free ;)