Everything I think you are likely to want to know is in this document. Sorry but I don't think it can be adequately described in the 2000 words one gets here.
http://www.ukba.homeoffice.gov.uk/sitecontent/ documents/visitingtheuk/coaguidance.pdf
The basics are:
The non-national is very likely to need a Certificate of Approvel (COA) from the Home Office.
Section 1 deals with the situation if the non-national is abroad currently.
If the person is coming in to get married then the couple are leaving permanently within 6 months, you apply for a fiance visitor entry clearance at a cost at �63.
If the couple are planning to stay longer, the application is for a fiance entry clearance and the cost is �500.
Section 2 and 3 deals with the situation if the non-national is already in the UK. I agree that understanding the rules is complex. It seems to pivot on whether the person has settled status or not - for which I think another word is "Leave to Remain in the UK". There are different categories of this - from "Permanent Leave to Remain in the UK" to time-bounded Leave of say 2 years. One can make an application to gain Leave to Remain in the UK if one is an overstayer to the UK and one has been living here for a long period of time. The application cost is �295. If successful, getting hold of the COA to get married is then straightforward.
What precisely is the situation?
How long in the UK?
Children in the UK?