I am running power through a conduit to a detached shop, 80 feet of wire, a 60 amp panel. What size and type of 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.