Miner is your physical mining device. Usually, it is a specialized ASIC miner (device with just one purpose to mine cryptocurrencies) or a PC with powerful GPU sufficient for mining. There are different types of mining hardware for each cryptocurrency depending on its mining algorithm.
Worker is a name for your mining device that you use as a login for your mining software. We recommend giving a designated worker name to every mining device. That way you can track down a faulty miner easily just by looking at the monitoring section of your profile page.
You can also run all of your miners under a single worker name and everything will be just fine. The downside of this approach is that if one of your miners does not work correctly, you will see a drop in your hash rate but you will not be able to determine which device is not running optimally.