Showerhead leaking when off implies many possible problems. It could be just one or several of them causing the leak. In the simplest sense, as far as the plumbing mechanism goes, water accumulated in the stem and at the showerhead will keep leaking even when you have turned it off.
Also, if the diverter valve that stops water flowing to the shower head when turned off is malfunctioning, then you will have water leaking through the showerhead. Whatever is the cause the problem is irritating, it leads to wastage of water and can spike your water bills.
Check for a Clogged Showed Head
Over time, the showerhead will get clogged, partially blocked by the accumulation of debris. Impurities in water, dust, and dirt will result in partial clogging which may not become obvious when the water outflow is at its optimal pressure but turning off the shower will lead to a reduced pressure of outflow through the stem and that will lead to water buildup. This water buildup in the stem and the showerhead will gradually drip out through the head. The leak may very well be slight dripping but it will be constant as long as there is some water in the showerhead and the neck of the shower.
Clean the Shower Head
You can uninstall it and use an old toothbrush to get rid of the clog. Rub the toothbrush, use some herbal cleaning agent if that helps or you can use vinegar. Vinegar works well to get rid of any lime buildup. Vinegar will also get rid of mineral deposits. If the showerhead has corroded and appears to be deformed to an extent that the pores have gotten clogged by the deformation of the metal, then you must replace the showerhead.
Replace Rubber Washer
There is an O ring which is the washer that can suffer wear and tear over time. This will lead to leaks. You may have a one handle or two handle shower. Both types will have washers. Always go for exactly the type of washers the manufacturer had in the original fixture.
The other two problems causing a shower head leaking when off are malfunctioning diverter valve and malfunctioning cartridge valve. Both need to be replaced. If you are unsure whether or not you can do it on your own, hire a plumber.