Red-breasted Nuthatches will search trees looking for food that may be hiding under flakes of bark, and etc. They will go up, down, and sometimes sideways on the branches and trunks of the tree looking for insects. They may at times catch flying insects too.
In the fall and winter months they will come to feeders looking for sunflower seeds, unsalted peanuts, and suet. They will hide seeds at times to find and eat later when food is hard to find. If their food is too large for them to eat with just one bite they will use their bill and hammer the large piece under the bark in a tree to hold it so that they can break it down to a size that they can eat.
And they can also use a rock to help hold the food for them. That is what this nuthatch is doing in the pictures.