I have just bought this game for the PC using Steam. I created a character and start a new world. My character already has an axe so I tried to chop down trees with it, however it won't do anything. It keeps making the chopping noise and no matter how long I keep chopping it wont take the tree down. Please explain to me what I need to do to make my game work.
I'm just going to sum up everything that has been said into one answer:
Make sure you are equipping your axe. Click on one of the tree trunk's blocks (preferably the bottom one) and continue to hold down the mouse button while still pointing the cursor at the same spot on the tree. The axe should swing around in circles, making a chopping noise for a few seconds until the tree falls down.
The comments to your question offer important advice: you must chop the same section of tree, and chop it over and over, to cut down the tree. A copper axe is an efficient way to harvest wood. Note that it actually takes 15 chops, regardless of tree size, to cut down a tree. Each block of tree yields one wood, though, so cutting down taller trees is more effective.
Getting better tools should be one of your early goals. Enjoy!