I'm going to start by saying that I've tried to look this up online, but I'm not knowledgeable about trees and so I came up empty handed.
This limb just fell off of this tree in the front yard. It's a very old tree, and sustained damage last winter from an ice storm. Can anyone identify the white cottony stuff on the inside of the branch? It's it something we need to worry about infecting the entire tree, or is it more likely to be something that "got in" after a crack from the ice storm.
The white cottony stuff is fungal growth - whether that's a problem in just this branch, or whether it's an issue for the heartwood of the tree is impossible to say. There's no evidence from your images of bracket fungus on the trunk anywhere, which is good because its presence would mean certain death of the tree over time. I can see that this piece of branch seems to have torn away from a large branch,the remains of which are still attached to the tree.
Given that the fungal mycelium is within the central wood of that branch, and is quite obvious at the broken off end, it may well be that the rest of that branch and possibly the interior of the tree are also affected. What you need is an arborist to come out and examine the tree, not least to make sure that other large branches are not likely to drop down suddenly and cause a serious accident, but also to ascertain whether the remains of the branch should be removed, and check whether the whole tree is compromised.
ID would be tough not being able to touch and test. There is a good solution though so you can make sure your tree is healthy and won't throw another branch at someone or somebody's car...grins. That would be to check out your local Cooperative Extension Service that a big University hosts. You could drive that branch over to the closest annex for ID and advice. These guys are the BEST to get landscape, planting, composting, preserving...ideas for YOUR area! For free or dirt cheap!
I am unable to tell what tree this is! Please send closeups of the leaves and bark. I would also be asking about an expert pruner. This must be an oak of some kind, maybe? The angles between trunk and limb look wide which means those limbs are strong like an oak tree's limbs. I'd be interested in reducing the mass and weight at the end of those limbs so the next wind storm won't break off more branches. I'd also get the grass peeled away from the bottom of this tree's trunk. Seeing that the area is too narrow for a 'circle' I'd just get rid of the grass on that end of the bed altogether. Use a bark to cover soil. Pull any soil or mulch away from the bark of that trunk. In fact, I'd love to see a picture of the trunk/soil connection.
