It's not fully understood how speech is produced, but it's known to be a complicated process in several stages. The first stage is generally lumped together in what's called the premotor processes (not the muscles moving things, but the thought process).
- There's the idea, the thoughts that the individual wants to convey.
- Then comes the word finding, which are the words the individual wants to use to convey those thoughts.
- Then the brain assembles the sounds that are necessary to make each of those words. 
Then in the second stage, in what we will call the mechanical process, the brain sends the signals to make those sounds in that particular order at that particular speed (in which you thought of) to the muscles that control the vocal chords, lips, tongue, and breathing apparatus.  To top it off, are about 40,000 neuromuscular events per second of speech. And in order to make a sound, the coordination of over 100 muscles needs to happen. These coordinated movements have to perform without a single flaw for fluent speech to occur. The movements required for speech are performed naturally without conscious control and they are monitored by hearing and touch.  These movements involve:
So how is speech affected in a person who stutters?
People who stutter do so because of a miscommunication between their brain and the mechanics of speaking (respiration, phonation, and articulation). They do not have problems with the premotor process.  This miscommunication happens at the moment of speech. So you know what you want to say, you just physically can't. 
Remember that stuttering is everywhere. It can be found in all cultures and races. It effects both men and women, and it's indiscriminant of how smart you are, how much you make, or what you do for a living. 
 Dennis Drayna, Ph.D., "Telephone Interview with Dr. Drayna, Chief, Section on Systems Biology of Communication Disorders at the National Institutes of Health", November 24, 2009.
 Logan, The Three Dimensions of Stuttering: Neurology, Behaviour and Emotion, 7.
 "Speech-Language Pathology/Stuttering - Wikibooks, Collection of Open-content Textbooks", n.d., http://en.wikibooks.org/wiki/Speech-Language_Pathology/Stuttering.
 Drayna, Ph.D., "Telephone Interview with Dr. Drayna, Chief, Section on Systems Biology of Communication Disorders at the National Institutes of Health."
 "Speech-Language Pathology/Stuttering - Wikibooks, Collection of Open-content Textbooks."
 Guitar, Stuttering: An Integrated Approach to Its Nature and Treatment, 5.