A single layer of planked boards (not cross-plied, the pertersfinger shield has been confirmed to not have been multiple layers) of any one of a number of species of tree (lindenwood is the common breed named in poetry, poplar and ash are common from archaeology). The boards were likeley to have been thicker in the middle and thin towards the end.
This would have been faced with hide (law codes exist expressly banning the use of sheep hide so we would expect cowhide was used). This is likely to extend pass the edge and be wrapped over to form a rim. There are some small clips found in graves (not universal) these may have been there to reinforce edges, cover gaps in the edge or just look cool.
A boss or umbo would be mounted with 5 clenched nails/roves (4 or 6 are possibilities but most of the examples i know are 5) in the center with a hole cut behind. There would be a grip or either wood or metal & wood through the middle of this hole (some metal grips are very long, most are short, wooden ones don't tend to survive), the shield was held in the hand by this (no forearm straps), like this guy.
A long strap on the back wouldn't have been uncommon (he has it in his hand), there are special fittings found in archaeology and the sagas mention them. This would presumably have been for carrying it around.
The Anglo-Saxon Shied by I.P. Stephenson