My wife and I are travelling to the UK. We are visiting a friend staying over for three nights. We then leave for Sweden for twelve days coming back to the UK via Scotland for 2nights. Do we need a multiple entry visa or do we just fill out the standard visa application?
The UK Standard Visitor visa is always issued as a multiple entry visa valid for 6 months, unless UKVI see a compelling reason to curtail the number of entries or validity period. Just provide the dates you plan to be in the UK in the application.
Edit: When completing the application I would provide the initial date of arrival and the final exit date in the ’Travel information’ section of the application. In the ‘Accommodation details’ section you can then enter your specific UK travel plan dates, and the address(es) where you’ll be staying if known. You can also use the ‘Extra information’ section at the end of the application to explain your plan to visit Sweden from the UK.
I'm guessing that you do not have a European passport and that you do not have visa exemption from either of these countries. In that case you will need a visa for the UK and another (Schengen) visa for Sweden. You're flying direct between the UK and Sweden I assume?
The UK has never been a part of the Schengen scheme.