Concrete walkways are a good solution when the walkway will be heavily used, as they are very durable. This is why most city sidewalks are built from concrete. On the downside, concrete is labor intensive and more complex to install than other DIY walkway options. It is also less flexible from an aesthetic point of view, as it is poured in place (although it can be stamped with a surface pattern). As it tends to dominate a landscape, concrete is fairly uncommon in backyards.
Even less aesthetically flexible than concrete, asphalt is also used largely for high-traffic locations. It will have the gray/black color of a roadway and be similarly hot in the sun. That said, asphalt can be a good choice for very long walkways; once the proper pouring and rolling equipment is brought in, it is easier to lay down than concrete (which requires wooden molds and pouring).
Interlocking pavers are a common walkway choice and have much to recommend them. They are made either from concrete or stone and are available in a variety of colors, patterns, sizes and textures. It is easy to match paver walkways to the look of the landscape around it, and they are relatively simple to install. If installed properly pavers are also durable, and individual stones can always be popped out and replaced if damaged.
Flagstones are flat pieces of natural stone chipped from larger formations and laid down to form walkways. They add a natural look to a walkway, as each flagstone is differently shaped. They are quite durable and age nicely in place. They are quite simple to lay down, although the individual stones are often large and heavy. A flagstone walkway will also not be perfectly flat, which may present accessibility issues.
A very simple walkway solution is to dig out the path of the walkway and fill it with a load of gravel or loose stones. The gravel is quarried at gravel pits, while the stones often come smoothed and rounded from rivers. They are very easy to put in; all you need to do is dump them in place. Both stones and gravel are available in a wide variety of diameters---the smaller the stones, the more even the walkway surface.
Ground-up tree bark and wood (collectively called "mulch") are soft and natural-looking choices for a walkway. Mulch can be laid right on the ground, but will periodically require replenishment as rain and rot dissolve the pieces of organic matter in the mulch.
Shredded rubber walkways use leftover rubber that has been torn into small pieces as a synthetic replacement for mulch. Rubber can replicate the soft feeling of walking on mulch, while not rotting away in the same way as an organic material would.
A more expensive alternative to stone, gravel or organic walkways is to build a boardwalk. These are essentially small decks with planks laid across a frame of wooden joists and secured with screws. They can be raised above the landscape or placed in holes dug deep enough to make the boards sit flush with the garden around them.