If you're only gonna unwrap the cylinder for UV-mapping all you will need is two seams; Mark the top face (in edit-mode), hit CTRL+E -> Mark Seam. What this does is, it creates an outline to the edges around the selected face. Now, do the same thing to the bottom face.
Once you have your seams, select every vertex of the cylinder and hit U -> Unwrap. Now you have created a UV-map. If you enter the UV/Image-Editor you will see the map created there for you, ready to be tweaked. However, since this is such an easy geometrical figure to create easy and ready to use UV-maps from I suggest you leave the map alone for now.
Next step is to make sure the texture is shown thanks to the UV-map. Different purposes require different techniques. UV-maps in Cycles require some node-usage. Here is a series of tutorials for this purpose (as I am not particularly good at explaining this procedure).
As for the Game Engine there are no requirements to use nodes, nor is it complicated to apply this UV-map to your asset at all.
First off you go to the materials window, add your material, go to the textures window and load up the texture of your choice )in this case; bark of a tree). Now, you scroll down the list in the texture-window until you see the tab; "Mapping". Within this tab, look for "Coordinates" and change this to UV (UV is the standard choice after the 2.72 update).
Now your texture will be visible in textured mode in the 3D viewport. However, it might not look as good as you hoped for right now so lets fix that. Select your mesh, enter edit mode and select everything. Then go to the UV/Image-Editor window I mentioned earlier and select the sides of the cylinder. If unwrapped correctly, it should be a rectangular shape). Next step is to move it around, scale it, twist and turn it to your liking. Moving the vertices in the UV/Image-Editor window does not affect the topology of the mesh, so don't worry.
When you're satisfied with the look of the textured mesh you can exit edit mode and add it to your game
if there's anything of this you don't understand, don't hesitate in asking and I will clarify what I mean!