Settling in the UK
Depending on the type of visa that you have, it may be possible to 'settle' in the UK (also known as 'indefinite leave to remain' or 'Settled Status' for qualifying EEA/Swiss nationals).
The ability to settle usually occurs after 5 or 10 years residence (depending on the visa you have) and requires you to meet other eligibility requirements such as knowledge of English language, passing a life in the UK test and not having excessive absences from the UK over the life time of your visa. Details on eligibility can be found here. You may be able to qualify to settle in the UK faster in some cases, depending on the type of visa that you have.
Being settled in the UK means that there is no time limit on your right to live and work here and it can lead to naturalisation as a British Citizen. While settling s regarded as an indefinite / permanent right to be in the UK, it can be lost in some cases. Full details can be found here.
To qualify for British citizenship, you must normally have lived in the UK for at least 5 years and have had either indefinite leave to remain or Settled Status for at least 12 months. You may be able to qualify for British citizenship quicker in some cases (for example if you are married to a British citizen). Full details on eligibility and fees can be found here.
If your application to naturalise as a British citizen is successful, you will normally be invited to attend citizenship ceremony and then be issued with your citizenship certificate. You can then apply for a British passport. Further details can be found here.