I have low spots in my yard that I would like to slowly fill in. This isn't anything big enough to constitute a truck load of dirt or anything. What can I use (either dirt or manure) to fix these spots?
Dirt can mean pretty much anything, from good to poor quality soil.
Manure "generally" refers to a natural fertiliser and is good for working into soil, adds organic matter, helps open up the soil... It's not the best material for filling in low spots.
Personally I would use a 50-50 mix of "high quality" screened (¼inch/6.25mm sieved) top soil & compost for filling in low spots in a garden, or lawn.
If filling in low spots on a lawn, I would fill to a depth of no more than 1inch (25mm) at a time and would fill in no more than twice a year, once in early to mid Spring and once in early Autumn (Fall).
The benefit of "dirt" is that it won't settle as much as manure, so you won't need to keep adding to it. But, depending on what "dirt" you use, it may be lacking in organic matter and nutrients. If you're buying commercial bagged soil, check the label to see what it has in terms of organic matter and fertilizer.
The benefit of cow manure is that it will provide those areas with lots of organic matter and nutrients. Horse manure has the same benefit, but comes with the problem that it can contain a lot of weed seeds.
Depending on where you live, soil and/or manure might be free for the hauling. If you have the choice of poor quality soil and manure, take a little of both and mix them. In my case, I have lots of free manure and not much soil, so I tend to fill low spots with manure. When they settle I add a little more.