Water evaporating off the dayside and permanently condensing on the night side, resulting in a desiccated dayside, is one potential outcome, but only for a certain range of atmospheric density and composition and water content.
It actually takes surprisingly little atmosphere to distribute enough heat around a tidally-locked planet to ensure that liquid water can still exist on the night side. But even if you don't have that, a sufficient quantity of water will ensure that glaciers completely covering the night side will nevertheless creep back around into the light where their edges melt and re-introduce liquid water to the dayside.
As for what the weather will look like, the details are highly sensitive to things like exactly how much atmosphere there is, how fast the planet is rotating (equivalently, what its year length is), how much surface water there is, etc. In broad strokes: it is very easy to end up with either a single permanent cyclone over the sub-stellar point, or a pair of permanent counter-rotating cyclones on either side of the equator; it is also fairly typical to end up with high-altitude super-rotation of the atmosphere (similar to what we see on Venus), while surface-level prevailing winds tend to run from day to night along the equator, and from night to day over the poles.
Slightly more details can be found in, e.g., Simulations of the Atmospheres of Synchronously Rotating Terrestrial
Planets, or any of a surprisingly large number of related papers.