Norton's answer is basically correct, although I will note that a native speaker would be much more likely to say "I have" a headache (not "I am having" a headache).  

The phrase "I have" is much more common, and means either that you own something (it belongs to you) or it is with you. "I have a cat." "I have a good job." "I have your keys in my pocket." "I have his phone number."

The phrase "I am having" describes a temporary event or process. You can't be "having" keys, because having keys is not a process or a state. We often use the term "having" to describe the process of eating food: "I am having lunch." "I am having pizza."  We also use "having" to describe a temporary state of confusion or difficulty: "I'm having trouble opening the door." "I'm having a hard time hearing you."  This means that I don't usually have difficulty hearing, but I am currently, temporarily having difficulty.  (If you say "I have difficulty hearing you," this means that it is always hard for me to hear you, because you speak too quietly, or because I have a hearing problem).  

"I'm having a conversation" means the conversation is going on right now.  You can also use "having" to refer to a process that will take place in the future.  For instance, "I'm having surgery on Monday."  Again, you're describing a temporary process that will occur only on Monday, not something that you "have" all the time. "We are having a party tonight."

(To follow up on the headache example, native speakers usually use the phrase "I have" when talking about types of pain or discomfort, even though the pain may be temporary.  There's nothing wrong with saying "I'm having a headache," but most native speakers don't say it this way. It suggests that you regard the headache as a process or event, rather than a "thing" that you have.)

In English, dynamic verbs have simple forms which refer to the complete action (I worked, I work, I will work) and  progressive forms which refer to an incomplete action (I was working, I am working, I will be working).  

Stative verbs usually only have simple forms which refer to a state.  In general (but there are exceptions), the verb to be, the verb to have, verbs of thought, verbs of emotion, and verbs of sensation are stative verbs and usually only have simple forms (I was sick, I am sick, I will be sick, I believed the world is round, I believe the world is round, I will believe the world is round, I hated spiders, I hate spiders, I will hate spiders).

A grey area is a temporary state which is treated like an action (with a beginning and an end).  "I am feeling sick" is an example of treating a temporary state as an incomplete action (I started feeling sick this morning  I am still feeling sick but I will probably stop feeling sick tomorrow.).

"To have" has multiple meanings: some dynamic, some static.

Dynamic - to have as a synonym for to eat, to drink, to meet, to telephone.  Example:  I had breakfast yesterday at 7.  I am having breakfast.  I had a meeting with Joe yesterday.  I am having a meeting with Joe.

Static - to have as a synonym for to own and to suffer pain.  Example:  I have a cat.  (*I am having a cat - not correct).  I have a headache.  (*I am having a headache - not correct)