You need to create a cylinder yourself in a 3D modeling program, and set its UV coordinates correctly using a UV Mapper, so that your texture appears correctly on the cylinder.
Unity comes with some basic primitive shapes (including cylinders), however their UV texture coordinates are also set to the default, thereby making your texture look like this.
So if you want to make your texture appear correctly, you need to make your own cylinder in a 3D modeling application, including setting up its UV coordinates properly.
Edit: You could also just use a plane if you don't need the depth... make sure to use a Transparent shader so the edges of your texture don't show. This won't work for all situations, though, mostly just 2D.