Electrical power to a detached shop, 60 amp panel, 80 feet of wire, what size wire do I need?
If the entire run is in conduit, you need individual wires rated THWN, THWN-2, or XHHW-2.   

The size of wire is 6 AWG in copper or 4 AWG in aluminum.  Either one is rated 55A but you are allowed to round up.   Also you are allowed to use bigger wire than strictly needed; if you use #3 copper or #1 aluminum you could increase amperage later to 100A.

The panel is required to have a shutoff switch because it is in an outbuilding.   The cheapest way to get a shutoff switch is use a panel with a main breaker.  The main breaker can be any size, since it's only serving as a shutoff switch.  

When your subpanel has a main breaker, that means the subpanel can be any size.  We strongly recommend a significant number of spaces, many more than you immediately expect to use. Spaces are dirt cheap when you are initially installing the panel, but it is rather pricey to replace a too-small panel.  Panels with more spaces tend to be larger than 60A, but it is fine to have a 60A breaker feed a 125A subpanel.  It can't do any harm.