In a school that I used to work in there was a kid who had a very bad stammer. It was a problem for him because he was more or less rejected by the other kids who did not have the patience to let him get his words out.

The official description of stammering is "characterised by stoppages and disruptions in fluency which interrupt the smooth flow and timing of speech. These stoppages may take the form of repetitions of sounds, syllables or words, or of prolongations of sounds so that words seem to be stretched out, and can involve silent blocking of the airflow of speech when no sound is heard" Speech may sound forced, tense or jerky. People who stammer may avoid certain words or situations which they know will cause them difficulty.
Some people avoid and substitute words to such an extent that people in their lives may not realise they have a stammer. This is known as "covert stammering".

Interesting topic! It is not known what causes stammering but research seems to suggest that a combination of factors is involved.
Genetics are relevant at least in some cases. Someone with stammering in the family seems more likely to develop a stammer themselves.
Brain imaging studies have shown significant differences between the brain activity of people stammering as compared with fluent speakers.
Stammering affects people in different ways and can vary according to the situation in which the person finds themselves: to whom the person is talking; how they are feeling about themselves and their speech; and what they want to say. Stammering can vary from adult to adult and child to child in its manner, frequency and severity.

Stammering is not simply a speech difficulty but is a serious communication problem. For the child or adult who stammers it can undermine their confidence and self-esteem, and affect their interactions with others as well as their education and even their employment prospects.

Various factors have an effect on the ease or difficulty with which people who stammer can speak. These can include:
A child or adult who stammers may stammer more when increased demands are made of the person in speaking situations, when the person has high expectations of him or herself in certain situations and with certain people (e.g. speaking on the telephone) or when a specific response is needed (e.g. saying one's name, address or phone number, having to use particular words) . On the other hand, in some people this stress actually increases fluency.
Children or adults who stammer do so on words which carry information and when using complex words of several syllables. They tend to stammer more at the start of sentences. Sometimes it is more difficult for people who stammer to speak fluently, for example when they are feeling ill, stressed, tired, excited, or upset.
Psychological factors:
People who stammer may stammer more depending on: their feelings about their speech,their perceptions of themselves as effective communicators and others' reactions to their stammering.

People who stammer are normally fluent when speaking in chorus, singing or whispering. (Weird that, but true, remember Gareth Gates who won the X factor or did he come second?)

Another interesting fact I came across is that it seems that some people seem to have a view that people who stammer are less intelligent, more nervous etc.

All research, however, indicates that the stammering and the non-stammering populations are identical in terms of intelligence, mental state, neurotic behaviour etc. The only difference found is that one group stammers and the other doesn't, but no explanation for this has as yet been identified.
Speed of speech should not be confused with speed of thought. A person who stammers will generally think at normal speed, even though expression of the thoughts into speech may be slower.

It is useful to know that the BSA Parental Awareness Campaign promoted the message that children showing early signs of stammering should be referred to a speech and language therapist (SLT) as early as possible.

Speech and language therapy has proved to be most effective with children aged under 5 years. In many cases when the problem is caught early enough the child is able to learn to speak fluently again with no evidence of recurrence.

For both older children and adults who stammer, the situation is more complex. Modern therapy can help improve fluency, confidence and communication skills but as stammering is more established by this stage it becomes more a case of learning to manage it effectively. As well as speech and language therapy many people find self-help groups useful in this regard.