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The need for a UK visa is determined by what passport the person is travelling on.
Having a residence permit in the US or Europe is not sufficient, because these are visas issued by other countries and are not recognized as travel documents. This includes permanent residence visas like Green Cards, European Permanent Residence Permits, and so on.
The UK keeps a list of which passport types require a visa and which do not, the list is periodically updated by Parliament and you can search for the relevant acts at HM's Stationery Office, but the most expedient way is to use the on-line wizard at Check if you need a UK visa site. This wizard program is kept in synch with the legislation and is never out of date.
If you are just passing through the UK on your way to a 3rd country, you need a transit visa if you do not qualify for one of the concessions. The need for a transit visa along with the various concessions are listed in the wizard cited above. There's a great article about the need for transit visas at Is there a way to find out if I need a transit visa for a layover in the UK?
For everything else, you need either...
As shown, this site deals with the Standard Visitor Visa only. These visas are appropriate for business trips, conferences, family visits, maintaining relationships, recreational courses, casual volunteering, shopping and tourism.
The rules governing the Standard Visitor Visa are at Immigration Rules, Appendix V.
The guidance about what sorts of paperwork they want to see included in an application is at Visitor: supporting documents guide.
The 'permitted activities' are listed in "Visitors Appendix 3, Permitted Activities for All Visitors and Visitors Appendix 4, Permitted Paid Engagements. These are found in Appendix V, cited above.
The list of permitted activities was hammered out back and forth between UKVI and the UK legal community and it is thought to be exhaustive. If you are proposing to do something that is not covered in this list, then you need some other kind of visa.
You can apply up to 90 days before your intended travel date.
There is no cooling off period following a refusal. You can reapply on the next day. However, note that UKVI will usually refuse only for applications that take a long time to fix. If the ECO thinks there is a quick fix, they will contact you and sort it out. If the ECO did not contact you, it means there's no quick fix and applying the next day invites risk.
You can apply to any British mission that accepts visa applications. You do not need to be in your 'home country' to apply for a Standard Visitor Visa.
Go to Visa4UK and open an account. Once you have filled in the basic information, the computer program will know what to do next. The final steps are to pay the fee and schedule an appointment at the nearest biometrics enrolment centre (the computer will show one or more alternatives).
The UK Government keeps statistics on how long it takes various nationalities to apply for a visa. You can access this information at Visa Processing Times.
A relative or friend can help, but if you are not completely sure of what to do, you can get legal help by selecting someone at the UK Law Society or the OISC.
You need a Standard Visitor Visa.