In many climates a firewood rack cover may be simply unnecessary, or even a hindrance to seasoning firewood. The theory behind using a firewood cover is simple - rain makes wood wet, so keeping the rain off must be a good thing.
Unfortunately this thinking doesn't make a distinction between surface moisture in wood, and the moisture bound up within the fibres and cells of the firewood. Wood that has a damp surface will usually dry off quickly in a good breeze, provided it is suitably stacked above the ground, and using a cover, such as a tarpaulin can actually slow overall drying of your firewood.
Some form of cover might be a good idea if your autumnal and winter climate is particularly wet, such as in the UK. If you are in a region where winter temperatures are consistently below freezing and winter precipitation is usually in the form of snow they may not be such a good idea
In sub-freezing weather firewood can continue to season, albeit more slowly, especially when stacked in an exposed place where it get both wind and sunlight. This is obvious in mountainous regions of europe where stacked firewood on open hillsides and field is very common. A firewood rack cover over these would actually be detrimental, as blocking the wind would slow down winter seasoning.
In very wet winters a cover can be great help as rain blown onto your wood stack may not have a chance to dry off properly before you need to use it. A firewood shed may be an advantage here - a fixed roof but open sides is ideal as it deflects the worst of the rain but, if well placed, still allows wind and sun to do its job.
The type of cover you choose will depend on what you are aiming to achieve. In a well stacked wood pile a simple plastic strip, weighted down on the top, may suffice to keep occasional showers and snow from settling. In more prolonged wet weather a full sized tarpaulin covering the whole pile and carefully weighted down may be necessary.
Some commercially manufactured firewood covers are available. The advantage of these is that they are usually designed to make the firewood more easily accessible without needing to remove the whole cover. This is a big plus, as a tarp that has been sitting out in the rain, even carefully arranged and folded, may collect water ready to dump down your legs when you come to collect wood.
Custom built covers may use velcro or elasticated cord to secure the cover to a frame. Others use weights stitched into the hems to ensure they hang in even strong winds.
Generally I would recommend that people shouldn't cover their firewood, atleast not with a tarp or similar heavy fabric. The consensus of people who have used them seems to be that they actually slow down the seasoning of wood.
The best use for a system like this is probably as a temporary rainshelter for firewood that is otherwise ready to burn.
We have, in the past, gone down the route of just flinging a tarp over a pile of partially seasoned wood and the results were disappointing to say the least. The tarp collected water in places and leaked in others and, even two years later, the firewood was only partially seasoned as it hadn't been exposed to the wind and dry air.
We now have our own purpose built firewood shed which is a much better solution.