Fellowships are grants made available to researchers to support both the costs of the research and the salary of the applicant. Fellowship schemes are available from the Research Councils, the Wellcome Trust, the Royal Society and several other charities. Although awards are available to researchers with established positions to allow them to be relieved of their teaching/admin duties, most awards are designed to promote the careers of postdoctoral scientists. The awards are made to the individual with the support of a sponsor (often not in the same institution) who will provide the lab space for the project. As such the award of a fellowship can be seen as the next major career step for a postdoc who wishes to pursue an academic career. Indeed, the award of a fellowship is likely to be the key to obtaining the elusive permanent contract. The total number of fellowships awarded is relatively small which ultimately means they are very competitive. So how best to obtain one? The first thing is you must have made some significant contributions that are recognised by your peers. You must also demonstrate 'independence' which is not always easy since your current supervisor must 'let go' and allow you to run with the project. The final thing is to ensure that you can perform well in an interview environment, be prepared for some very searching questions.
Despite these academic requirements, perhaps the most important thing is self-awareness. Do you really believe that you can be the lab head, the predominant intellectual/technological drive, the one who can sort the problems out when it all goes wrong? If you doubt that you can, then you probably won't be successful.