I consider the primary effect to be the incredibly limited habitable zone on the planet. This is now a thin strip where permanent dusk or dawn exists. To either side is a frozen wasteland and a desert wasteland.
Unless of course the inhabitants have done something about it. Given Earth, if this were to happen slowly, we could potentially create light-directing and diffusion structures. Combined with the right greenhouse gases, we could potentially normalize the temperature on the planet to some degree. This would have to occur over very long timescales.
Remedy or no, one side will be eternally night and the other eternally day. You'll have a number of strong weather phenomenon for sure in the beginning. Once wind starts being driven by a single source and direction of convection, large weather system will likely dominate. There's a possibility of heavy cyclonic activity in the habitable zone due to pressure and temperature contrast.
Subterranean water, oils and gasses will explode outward into the atmosphere as the desert side gets progressively baked. Meanwhile the night side's oceans will freeze, the atmosphere will get thinner as more than usual condensation falls from the sky and any rivers or oceans will follow the same fate. Inside the planet there will be a region that was previously too hot now entering a habitable scale of temperature, but its likely covered by tons of ice or rock.
Depending on the gasses spewing out of geysers on the day side, you might get a runaway greenhouse effect or maybe not.
Worst of all, if the planet becomes tidally locked its magnetosphere is very probably going to be screwed. Without a spinning iron/heavy metal core to generated this shell of protection, the atmosphere is going to be more easily blown away and cosmic ray radiation will start pouring in.