Shower Constantly Dripping
You have two issues.

The first is the dripping when the shower is off. Most likely your gaskets are worn and need to be replaced, though it could be another issue in the valves such as worn valve seats. These gaskets are rubber parts inside your hot and cold water valves (and possibly in your diverter valve) that prevent water flowing through. There are valve replacement kits available at hardware stores for many common brands. The process is a little different for different faucets though, so ideally you would identify the brand you've got and look for guidance on that specific brand.

Next the dripping when the shower is on. A few things this could be:


Dripping from where the shower head attaches to the shower arm. This is a sign of poor connection of the shower head, and can often be fixed by tightening. Only hand-tighten here because most shower heads are plastic and can break, and because you don't want to strip the threads on that shower arm. If that doesn't help, check that the shower head isn't already cracked, and check the gasket in the shower head that seals it to the arm.
Dripping from the face of the shower head when it's on. This is normal, as @isherwood commented. If it's severe then maybe some of the shower head's outlets have mineral blockages. Clean/soak with vinegar or CLR to unblock.
Dripping from the tub faucet after you've diverted water to the shower. Some dripping is probably normal depending on your shower fixture, most say a pencil-width stream is OK. If it's much more than that, or has gotten significantly worse over time, then the issue is likely worn gaskets in your diverter valve.


For more specific guidance, please identify your fixture brand and specific issue(s). Good luck!