What they are
The fascia is the board along the edge of the roof that the gutter is fastened to. The soffit closes the underside of the overhang. Both take the water a failing gutter puts in the wrong place.
How to check
From a ladder or with a long screwdriver: press along the bottom edge of the fascia behind the gutter, at corners and under seams. Wood that gives, crumbles or stays dark after dry weather is rotting. Aluminum wrap can hide a soft board; look at the ends and where the wrap has split.
Why it comes first
Gutter hangers screw through the fascia into the rafter tails. In rotten wood they pull out within a season. When gutters come down for replacement, the fascia is open to inspection and repair at the least extra cost, so a replacement quote should say how fascia repair will be priced.
Houses built before 1978
Paint applied before 1978 may contain lead. Under the EPA's rule (40 CFR 745), paid renovation work that disturbs more than 20 square feet of painted surface on the outside of such a home has to be done by a certified firm using lead-safe practices. Removing a run of painted fascia can cross that line. Your town's page gives the share of homes built before 1980.