I think of a hard-shell jacket as a literal shell that protects you against the elements. This type of outerwear is waterproof and windproof, with a hood that cinches down around your face and cuffs that can cinch down over your gloves.
Ideally your hard shell should be breathable, too, so that condensation within the jacket can escape at the same time rain and snow are held at bay -- but there is no such thing as the perfect balance between breathability and weatherproofing, which is why many hard shells still come with pit zips. You can open the zips for increased ventilation, or close them down for better protection in foul weather.
Breathable fabrics have advanced a lot in the last couple of decades, so sometimes you'll see hard shells without pit zips -- but the best garments will replace those zips with mesh-backed pockets that you can open as vents, or with strips of soft-shell fabric along your sides and under your armpits. The soft-shell fabric sacrifices a bit of weatherproofing in that relatively protected area of your body, in exchange for superior breathability.
If your hard shell has neither pit zips nor pocket vents nor soft-shell strips, you're going to have to regulate your layering and activity levels carefully to avoid ending up soaked in your own sweat.
Layering With Hard Shell Jackets
Hard-shell jackets don't have any built-in insulation -- they're just the weatherproofing that goes on top of your other layers. That makes them very versatile, because you can adjust your insulating layers to suit your activity levels. (Here's how to dress in layers.) Make sure you buy your hard shell large enough to fit a layer or two underneath; otherwise, you won't get full use out of it.
The inside of older hard-shell jackets used to feel clammy against your bare skin, but many modern hard shells are soft enough to be worn over nothing but a t-shirt if that's all the insulation you need.
The Right Weatherproofing
When you buy a hard shell, check the label to see just how waterproof and breathable it is. There's no single industry standard for measuring breathability, so take this number with a grain of salt -- but in general the higher the better, and it should be at least equal to the waterproofing rating.
When it comes to waterproofing, measurements are standardized and again, the higher the better. I suggest looking for a waterproof rating of at least 10,000mm at the bare minimum -- preferably 20,000mm so that it can better withstand long periods of wind-blown rain.
Hard Shell or Soft Shell?
If you do a lot of highly aerobic (fast, steep) hiking in mild weather, a highly breathable soft-shell jacket might be right for you. Ditto if you find you absolutely must have stretchy fabric for a good fit -- soft shells will stretch with you, while hard shells will not.
But if you know you'll face bad weather from time to time, I recommend sticking with a hard-shell jacket. They're usually lighter and more packable than soft-shell jackets too, and just all-around more versatile and better protection when the chips are really down.
And if there is room for just one jacket in your budget, I recommend making it a good hard shell. This is one case where you definitely should buy the best your money will allow. Invest now, so that you can benefit from solid weatherproofing -- which can be a matter of life and death -- for many years to come.