For anyone attempting to make a decision on which visa to make an application for to meet your UK – based British partner or spouse, I hope the explanation below is of assistance.
N.B. This has no application to any person who made an application for their spouse visa, fiancé visa, or an additional leave to stay before July 9, 2012 or for partners or spouses of non-UK (EEA) nationals. Those paths for visa are entirely different and simpler.
Please also take notice that I am not an expert. The following details are just information that I have compiled while undergoing the process personally. So you are still advised to go through the website on UK Immigrations and Visas for the official requisites.
If your spouse is already presently a British Citizen, then you can go ahead to submit an application for a Spouse Visa. After a period of two and a half years living in the UK, you can proceed to make an application for a Further Leave to Remain. Eventually, after a period of another two and a half years (which makes it a total of five years in the UK on the whole) you can make an application for an Indefinite Leave to Remain.
If on the other hand, your spouse is NOT yet a British Citizen, you must make a decision of where you would both be married. This would expose you to one of two available choices:
[A] If your marriage would be performed outside the United Kingdom, then after your marriage you can submit an application for a Spouse Visa. After a period of two and a half years residing in the UK, you can proceed to make an application for a Further Leave to Remain. Eventually, after a period of another two and half years (which makes it on the whole a total of five years of stay in the UK) you can make an application for an Indefinite Leave to Remain.
[B] If your marriage would be performed within the United Kingdom, you would have to determine if you would be resident in the United Kingdom after your marriage.
[B.1] If you choose to reside in the UK after your marriage, then you would be eligible to submit an application for Fiancé Visa. Then after your marriage, you can make an application for a Further Leave to Remain. Two and a half years later, you would be eligible to make an application for a second Further Leave to Remain. And eventually, after another period of two and a half years (which makes it on the whole a total of five years of stay in the UK) you can make an application for an Indefinite Leave to Remain.
[B.2] If you choose not to reside in the UK after your marriage, then you would need to make an application for Marriage Visit Visa. This would entitle you to reside in the United Kingdom for a period not exceeding six months, within which you are expected to get married.
From the above explanations, it can be inferred that apart from the applications for Further Leave to Remain and the Indefinite leave to Remain which can only be made from the UK (while the applicant is already there in the UK), all the other types of visas (Spouse Visa, Fiancé Visa, and Marriage Visit Visa) can only be made and submitted in the home country of the applicant and only by the applicant.
Update at May 18, 2014
For more details on each visa and their requirements for finance, visit the UK Immigration and Visas website (for the rules, documents required, and the application) and the UK Citizenship and Yankees Visas Forum (which is appropriate for understanding everything, getting questions answered and easing stress caused by visas). If you have enquiries, freely make your comments here or enquire from the UK Yankee community who are extremely knowledgeable.