Conventional Milling:
when the cutter is rotating in clock wise direction and work piece is fed from left to right is called conventional milling; meaning it looks like it is trying to pull the material out of the work piece. Generally conventional machining is used for roughing and hardened parts. (could be case hardened). In this method there is chances of material to pullout from the work holding devices.
Climb Milling:
climb milling seems opposite to the conventional Milling. The cutter is rotating in clock wise direction and work piece is fed from right to left. In my words, it looks like it is digging in to the material. In case, we are looking for better grip of the material while machining this is the preferable method.
Please check this Animation out for better understanding of the concept.