Welcome to PF, brian. Lenz's law is an interesting and useful shortcut for working out the direction of a force. The hand rule is the basic and longer way to work out many of these situations. There isn't much more to be said unless you are interested in a specific situation.
The classic demo for Lenz' law is dropping a strong magnet through a copper pipe. It falls s l o w l y. Lenz' law says the magnetism induces a current in the pipe which acts to oppose the motion of the magnet.
A similar apparatus has a piece of copper pipe suspended by threads so it is free to move. When you pull the magnet out, Lenz' law says the copper pipe tries to hold on to the magnet and you can see this because the magnet pulls back on the pipe and makes it move. Here is the explanation of that effect using the hand rule: