I'm thinking about commissioning a piece. What kinds of art media do you work with? What are their benefits and drawbacks?

To make this more manageable, I'm going to break this down into sections.

Traditional Media

graphite

Graphite can be used in sticks or drawing pencils, or ground into dust and applied with a paintbrush. It lends itself well to soft, expressive black and white pieces, as well as adding detail, shading, and structure to mixed media color pieces without looking out of place. It takes well to many different surfaces, requires no dry time, and can be corrected more easily than most media (to a point- it isn't infinitely erase-able). It does have some downsides, however. It can be difficult to achieve a strong black (and even more difficult to remove one once you have achieved it). It is susceptible to smudging, especially on smoother surfaces, and special care must be taken to keep the finished piece from becoming a blurry mess during transportation or storage.  Problems can also arise during digitizing and reproduction, since graphite tends to be shiny and subtle nuances of shade can be lost in the process.

ink

I love working with ink! It is so much more versatile than its reputation for indelibility might lead you to believe. Its used in ballpoint, micron, and technical pens, with nibbed dip pens, or painted with brushes. It is easy to get a very bold, graphic effect with ink, as it is often quite literally working in 'BLACK' and 'WHITE' rather than the 'shades of grey' of graphite, but there are also many techniques to achieve the range of values in between, whether through using small, precise marks in varying densities (like stippling), or by watering down the ink itself and painting transparent washes (much like watercolor)- this allows ink to achieve gorgeously nuanced, delicate effects. There are also many colors of ink available beyond basic black. Ink is inexpensive, tends to hold up well over time, can be easily corrected in black and white pieces ('bleedproof white' is a wonderful wonderful thing...in reproductions), is can usually be digitized and reproduced with the most minimal effort, and tends to hold up clearly when reduced in size. There are a couple of downsides, however- it doesn't always lend itself well to linework in mixed media (it is wonderful for some styles, but it can also easily become jarring or overpowering), and those beautifully clear, clean lines can very easily come to resemble something splatted on your windshield if you don't keep careful track of both hands while working. Also, when working with a dip pen, the ink well can become something of a Sword of Damocles, poised to be knocked onto everything in the immediate vicinity and scheming to take particular aim for your artwork and your self.

So, in summary, ink:
pros- almost everything
cons- plotting

marker

At the time of writing this, I am not very proficient with markers but intend to put in the time to change that. They are applied in layered transparencies. They can be tricky to blend or mix, but offer a lot in achieving controlled color detail. The alcohol based markers do NOT play well with many ink pieces (will dissolve and smear the ink unless applied first).

colored pencil

These are applied in semi-transparent layers, and like markers also offer a lot of control in creating small color details and textures. They are fairly blendable, and work well with many other media. The effects possible with them depend largely on the surface they are being applied to; very smooth surfaced don't allow many layers to be put down, and toothier surfaces that hold more pencil wax often maintain that textured, stereotypical 'colored pencil' texture that might not be desired in the final piece. Colored pencil can also smudge, and can even melt if transported or stored in the heat.

crayons

Most of my childhood refrigerator art is in crayon; at the time I found them infinitely preferable to colored pencils. Some were scented or glittery, which was pretty rad. I don't use these anymore.

watercolor

 Watercolor paint is wonderful for creating pieces with a lot of expression, ambiance, and personality. Because it it is quite literally working with colored water, it lends itself very well to fluid, organic effects. It is well suited to creating artwork with a loose, gestural quality, although fine detail can definitely still be achieved- this makes it a standard in many field sketching kits. Watercolor artwork tends to have a luminous quality, since it is created by layering transparent color washes that still allow light to bounce off of the white paper underneath.  Watercolor can be tricky. It's lovely when it goes well, but when it starts going wrong it can go VERY wrong and it is difficult to effectively correct major errors due to its transparent nature and the way that the pigment sinks into the paper. It can easily become muddy if overworked. Many other major watercolor errors are caused by impatience; many effects require that the paper have exactly the right level of dampness (frequently requiring nothing beyond a soft sheen and coolness to the touch), adjacent areas will bleed together if they haven't been allowed to dry completely before continuing, etc. The current humidity level can have a significant effect on how quickly a piece can be finished (or how much time there is to use a technique). Using too much water can also cause problems, from watermarks to warping. It generally works best when used on heavy specialty papers, which can be fairly expensive (but are worth the investment for the part that they play in producing a quality final piece).

gouache

Gouache paint is one of my absolute favorite mediums to work with. It is also far and away the most expensive one, due to the tiny tube sizes and the amount used in each piece. This figures. It's still incredible. Gouache is considered an 'opaque' medium, although individual pigments vary in translucency. Vibrant colors. Mixing it with ox gall or water makes it wonderfully liquid smooth to apply to your surface (mixing in more allows you to layer it in transparencies like watercolor), it blends easily in most stages of drying, layers well to a considerable extent, and mistakes can be easily covered or sometimes melted off with a damp brush. The downfalls are that it is very sensitive to water, so the finished piece can melt if it gets wet, and it can crack if it is painted on a surface that gets bent or flexed (so rigid illustration boards or heavy watercolor papers are ideal).

acrylic

Acrylic is a very versatile medium that can have a bit of a bad reputation. It's cheap (although, like with all mediums, higher quality costs more), easily found, will 'take' to a very wide variety of surfaces, and comes in an incredible diversity of colors. It is opaque, plastic-based, fast drying, water soluble when wet, water repellent when dry. These qualities allow it to be watered down into washes and used in transparent layers like watercolor, mixed with gels and other assorted media for a variety of effects, or used straight. Its opaqueness makes it very forgiving of mistakes, which can be easily covered up with more layers of paint; its fast drying time can contribute to more mistakes in need of covering up, however. A thick layer of acrylic when dried will tend to create a plastic-y, water-resistant surface that doesn't lend itself well to further washes or the use of other media. Its uniquely bright, saturated colors lend themselves well to bold, graphic pieces (although it is not at all incompatible with more subtle work), it is a staple in mixed media pieces, and its drying time allows the artist to work significantly faster than with most other paints. 

oil

The defining word for oil painting is 'control'- this medium allows an unparalleled amount of it. This is the medium of the Old Masters. The rich pigments are mixed with oils for varying degrees of opacity, applied layer upon layer to canvas. It is slow-drying, which allows for a great deal of leeway in making modifications and corrections once it has been applied, compared to the 'wait for it to dry and cover it up' approach needed with acrylic painting, but its drying speed also very much sets the pace for how quickly a piece can be completed. Painting with oils also involve some potent fumes, so decent ventilation s necessary.

digital media

Digital media offers some advantages over traditional, namely the 'undo' button. It allows for much greater control, especially the ability to make significant changes to the overall piece without having to start over (such as changing a color, shape, rearranging elements to create a new composition, etc). However, care must be taken to prevent digital pieces from looking too 'plastic' or cartoony or flat- often, specialized brushes or textures must be used to emulate some of the character and visual interest that naturally happens in the creation of traditional media pieces.

raster graphics

Digital images of this kind are pixel-based. Because these images are ultimately comprised of tiny dots that form a coherent whole, they can only be enlarged so much before the photo quality suffers and they begin to look grainy. High resolutions look more refined and allow greater leeway in sizing, but the price is a much larger file size. Once shrunk, rasterized images lose information as pixels remain the same size, and many are discarded to maintain the look of the image at its smaller size. The benefit of raster graphics editors is that they tend to allow much more subtle, varied, and detailed effects than vector graphics. Adobe Photoshop uses raster graphics.

vector graphics

Vector graphics store information mathematically, rather than on a pixel by pixel basis. Lines, curves, shapes, sizes, colors, simple gradients... all are defined and stored numerically, which not only keeps file sizes low but also allows images to be enlarged indefinitely without losing any sharpness. This sharpness is both a strength and weakness of vector graphics- generally, it is best suited to typography, logos, cartoons, anything that doesn't suffer from a bold, graphic look. Adobe Illustrator is a vector graphics based program.

the final verdict

Fortunately, non of these media have to be used exclusively! Mixed media is a completely viable way to bolster the weaknesses of one medium with the strengths of another. Traditional media can be mixed for a wide variety of effects- ink and watercolor, colored pencil and acrylic and gouache, or combined with digital, whether painted on a printed base or making significant adjustments and corrections to an almost-finished piece. Digital pieces can be created using both raster and vector graphics editors, with files imported from one program to the other, depending on which form is more conducive to a particular stage in the illustration process. I personally prefer to create linework for pieces using a vector graphics editor, where individual components can be scaled up or down and rearranged until I have the composition that I'm looking for, then either imported into a raster graphics editor for 'painting' or printed and transferred to the canvas for a more traditional piece.

So, ultimately the best medium to use depends entirely on what you want from your finished illustration. Having a clear idea of the overall mood or effect that you want to achieve- bold and graphic, subtle and delicate, ornately detailed, pared down simplicity, hyper-realistic, cartoony, luminous and ethereal, earthy rich colors, etc; all of these things can inform what the best choices would be for your final artwork.