Termite mud tubes (or simply, termite tubes) are a very common mode by which termites use to travel across exposed terrain. The pictures below show what they look like. These tubes are frequently made of mud/dirt and/or termite droppings, and are one of the common signs of termites, especially in places where termites are found in abundance.
The purpose of such tubes is to provide shelter and protection to the termites from marauding predators, since the termites can travel unseen. They also minimize moisture loss among the termites traveling to and fro along the tubes (many termite species don’t survive long if exposed to the air). That said, not all termite species build such shelter tubes, and there are quite a number that are comfortable foraging in the open air and on exposed ground, although these are not your usual “destructive” termites.
In tropical forests, termite tubes can frequently be seen snaking their way up and down many trees; a large percentage of the trees would have these tubes on their bark. Often, there is a nest among the branches of the trees or on the trunk, and these tubes serve to connect the termites to the ground. Nasute termites are the ones that frequently build such tubes seen in the forests, although their tubes tend to be made of carton type material, not mud. They can even build these tubes at ground level, across paved roads near to forested areas that have not been traveled on for a long time.

Above – An old termite tube covered in moss, on a rainforest tree. These are constructed by nasute termites and made of carton material, so they can last a long time. This one has lichen already growing on it.
You don’t have to go into the forest though, to spot these tubes. Some trees outside my house such as the one pictured below, also have such tubes on them. These are constructed by Coptotermes termites. Although you might think the tree will soon perish, it may still take years for the tree to die, if ever, simply because the tree might just only be a secondary food source to that particular colony, or a mishap might happen to the colony in the meantime.

Above – A termite mud tube on a tree outside my home. It’s rather well camouflaged, and originates from crevices within the bark, which is inhabited by the subterranean termites.
Subterranean termites may build their tubes from the crevices found within concrete walls. Concrete houses will always have gaps within the concrete and bricks, which are easily exploited by subterranean termites. In heavy infestations, when a lamp socket or cabinet is removed, you may be able to observe these termite mud concretions behind or within them.

Above – In homes with termite infestations, remains of termite “pavements” can be seen within the walls (when the walls are demolished). These are not exactly mud tubes, but similar. Termites always pave or cover their trails where possible.
Termites can also build mud tubes in empty, open space to connect them to the ground, although not all species do this. So, a floor raised above ground may not offer foolproof protection! If you are looking for termite mud tubes in your home, check the zones that are near to, or connected to the ground; these are the places that most frequently have them.

Above – A typical termite mud tube.

Above – The bases of termite mud tubes near the floor are usually thicker and larger than the tubes themselves, and teeming with termites.

Above – A termite tube is seen coming out from crevices in the wall. Subterranean termites can exploit wall crevices very well.

Above – Termite mud tubes can traverse up the ceiling too, as in this case. Where possible, they follow the wall contours.
One thing to note; termites do not always frequent or populate their tubes, so empty tubes do not mean there are no termites in the vicinity. Termites tend to forage more under wet/moist conditions, so they are more likely to populate their tubes after rainy weather, and vice versa during dry weather. They will abandon their tubes in a particular area, once the food source there has been used up.
When termite mud tubes are found in a house, it usually means the infestation is already well underway, and may already be serious, because termites only build these tubes when they have to – As when there is no way to get to their food sources (the wood in your house) via normal means (the ground), and usually when the colony is sufficiently large enough to warrant the risk.
Otherwise, they would just quietly work their way up from the ground underneath your home. If you spot termite mud tubes in your home, it’s best to consult a pest control operator. Destroying the tubes will do nothing to make the termites go away, but may alarm and compel them to shift their activity somewhere else (in your home) if they do not rebuild back the tubes.