What fuel (burning material) gives the best flavor to meat when barbequing?
There are two main types of charcoal, briquettes and hard lump. The first comes in preformed pieces, so they are all the same size and include a filler material. They will sometimes also have lighter fluid in them already and be "easy start" or something similar. Hard lump is just charcoal from hardwoods with no filler. These will usually be in all kinds of shapes and look like "natural" pieces of wood (albeit in charcoal form). Hard lump is easy to light, burns hotter, and leaves much less ash than briquettes. It also has a better flavor. I strongly recommend hard lump charcoal.

Wood is usually used to provide smoke as a primary function rather than heat. Smoking on your grill gives a lovely smokey flavor (d'uh). The standard way to use wood is to soak it in water for a period of time prior to cooking. Packages will often say to soak for an hour, and that's ok with wood chips which are very small (but I recommend longer), but it isn't good for anything larger. I generally soak wood for at least 18 hours. 

Wood also comes in various sizes, from very small chips (slightly larger than a coin) to large chunks bigger than a fist. The smaller the chips, the less heat they provide and the faster they burn up. These are generally better for shorter smoking times. Large chunks can smoke for 12+ hours if they are properly soaked. In both cases, I put the wood on to already burning coals to get them going. 

You can also use just wood to create your heat, in which case you do not soak the wood. When you hear about woodfire ovens, this is what they do. However, I think this is a hassle as it's harder to get the wood lit than charcoal and doesn't provide as much smoke since you haven't soaked the wood. So it's really sub-par in my view. The exception is if you are doing low and slow smoking on some types of charcoal grill, you can use a small bed of coals to light large chunks of soaked wood. Once they get smoldering, this becomes the only source of heat as the coals go out. They smolder hot enough to get new wood going if you find you need more. Great for 12 hour smoking at 200F for Boston Butt or Brisket. 

In terms of wood, you want hard woods with low amounts of resin / sap. No glue, nails, etc. It wants to be natural. This is a decent description of types of wood and the smoke they produce. Common woods for BBQ in the SE US are hickory, mesquite, and cedar. I've used old wine barrel chunks which have a really nice flavor, and also bits of rosemary branch (thick from a large plant) which didn't turn out as well as I'd hoped. You can even use the shells of nuts like pecans. This is really a question of seeing what's readily available and experimenting with your own personal taste. 

You can also use wood with a gas grill by putting it in a smoking box or tin foil with holes in it, and placing directly on the heating element. Wood chips work best for this. I've never had much luck with this though. It's hard to get the chips smoking and, more importantly, gas grills don't have the airflow through them to properly move smoke over the food. I switched to charcoal pretty quickly and a Big Green Egg about two years ago, and I've never looked back.