A fellowship can have several different definitions, depending upon the extent, what it offers, and what it requires. Most often, they are monetary awards — scholarships — connected to working in a specific field, usually at the graduate or post-graduate level. While former or continuing graduate students complete extra training in a field, or have funding to continue research, they usually receive a stipend, slightly above living wage, from either a private or public institution. This allows them to complete their training, internship or research without having to devote time to working outside their field, or to resorting to additional loans for support.
There are some very specific uses of the word fellowship in the medical field. Doctors who are completing training in advanced specialties, like oncology or pediatrics, are normally given a fellowship or stipend while they complete their training and work for a hospital, and they do indeed work hard during this training period. In the US, these specialists in training are referred to as fellows, though they can also just be called doctors since they have completed medical school and are licensed physicians. Their designation as fellows denotes that they are training in a medical specialty.
Other types of upper level training with pay may be called fellowships too. For instance, doctoral students who must devote time to teaching may be called fellows, and may be compensated for their work scholarships or fellowships. People training in the mental health industry might complete their training or further research, or their required "hours" prior to receiving certification with partial help from a fellowship.
Sometimes a fellowship is much more brief, and may consist of working or researching for a month or two at most. Museums may offer short ones to advanced students, primarily during the summer months. Various humanitarian agencies offer them as well, even to some undergraduate or non-students, in order to expand programs, fulfill missions, or allow a person access to learning or mentorship.
Since fellowships often involve working in your area of study, garnering one can kick-start a career and make an impressive addition to a resume. Working with professionals, receiving mentoring, and making connections in a chosen field often leads to job offers and better career opportunities in the future. It is often worthwhile for a person to explore opportunities in her chosen field to see if any exist for which she might qualify.
Qualification for these work/research scholarships varies. Some are dependent on income, but many are more dependent on applications and academic or career success in the past. Many programs are very selective and competitive, and such programs may help participants become working professionals in their fields.