Haben is the broadest term when you talk about having something. Hence, you will find various examples of usage for it. I will only give one:
Ich habe ein neues Auto.
(I have [got] a new car.)
Besitzen emphasizes that you have something at your disposal, i.e., you own or possess something:
Ich besitze zwei Autos.
(I own two cars.)
Innehaben is a higher-register term used when someone holds an office or occupies a good position, for example, at university or government. Used with things other than these, innehaben is considered stilted:
Er hat zurzeit den Chefposten inne.
(He currently holds the chief position.)
Ich habe eine Wohnung inne. (rather stilted)
(I own an appartment/ flat.)
Since it was asked for in the comments, I would also like to address the phrase über etwas verfügen. More than besitzen, it emphasizes your influence and power of disposition over something. I am not sure what the best English equivalent would be or if you would need more words to express the emphasis in English:
Ich verfüge über wichtige Kontakte.
(I have/ dispose of important contacts.)