Mars Water Cycle
This is the modern water cycle where water enters the air from the ice caps, and frosts out again some time later on the colder side of the planet.
Note that in usual Martian conditions, water ice does not melt (into a liquid) but rather sublimes directly into a gas.
If all the water on Mars was to melt, it would form a covering 20 to 30 meters deep over the whole planet (assuming the planet was flat). Thus Mars' water inventory is significantly less than Earth's (with oceans ~2 km deep).
Overview
More than 2/3 of the water in the Martian atmosphere comes from the north polar ice cap, a significant proportion from subliming frosts in the seasonal ice caps in the north and south poles, and some smaller amount from humidity trapped in surface soils. There is some disagreement on how much comes from the soils.
The water trapped in the ground & ice caps contains 1 to 10 million times more water than is in the air.
Mars currently is losing water at a rate of 10^-6 kg / m^2 / year. So the current Martian reserves could replenish the atmosphere for billions of years.
Mars is closest to the sun at southern winter, so the South Pole gets ~20C warmer than the North Pole during its summer. However, the norther summer is much longer, and the southern winter is much longer. So much so, that a significant amount of total CO2 atmosphere freezes in southern winter, forming an cap of solid CO2 8 meters thick. This traps the water ice and keeps it from subliming in southern winter. Thus the majority of water in the Martian air comes from the North Pole.
The water sublimes from the seasonal ice cap (frost and a bit of snow) in the northern spring and summer. This enters the atmosphere and is distributed world wide. Water clouds are only common in northern summer. Much of this water frosts out near the South Pole. With southern spring and summer, these frosts sublime entering the air, gradually frosting out (or forming snow) in the northern hemisphere.
The moisture can form clouds, and dust particles trapped in ice shells or snow are pulled out of the air. Thus the water cycle removes the smaller dust particles from the air.
Water is most common in the air in early northern summer, and is lowest in northern winter. The Southern Hemisphere has significantly drier air.
Seasonal Reserves
References
"The Atmosphere and Climate of Mars", Edited by Robert M. Haberle, et all, Cambridge Planetary Press, 978-1-107-01618-7.