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
In northern summer ~85% of all water in the air is in the Northern Hemisphere. In the southern summer, only 60% of the water is in the Southern Hemisphere. Global Climate Models have been run where they start with equal amounts of water ice on both poles, and they find that the water migrates to the north.
How high water reaches in the Martian atmosphere is not well understood, but it seems that most of the water is found in the bottom 3 km of the air.
It has been estimated that the Martian polar caps contain between 3.2 to 4.7 million cubic kilometres of ice. (Tho the South Pole might have more if it is deeply buried.) These polar caps form the main reserve of water accessible to the atmosphere. The polar ice caps are not pure ice of course, they are estimated to be contaminated by 2.5% dust. (In more equatorial regions, this increases to about 5%.)
The north polar cap covers regions higher than 80 degrees north. The south polar cap is smaller, only covering regions higher than 84 degrees south.
The southern ice is covered with 8 meters of CO2 during the winter. (There might also be significant amounts of solid carbon dioxide buried deeper at the South Pole.) This slows the water subliming in the short southern summer.
Additionally, water is in the soil as permafrost. This is ubiquitous at latitudes of 60 degrees or higher. The depth of water ice can range from mm to a several meters (in extreme conditions), more often it ranges from 10 to 100 cm, but is generally found 30 cm below the surface. It is closest to the surface near the poles, and deeper at the equator.
Another source of water is "adsorbed" water bound to soil particles. This is a thin layer of water molecules which attach to soil. They form a layer too thin to be ice, or frost. They easily are deposited at night and easily escape during the day as the soil warms. This accounts for the rise in humidity in the morning, even if the temperatures are not high enough to sublime ice.
Measuring Water in the Air Column
Water on Mars' air is measured in units of "um pr / m^2". This is micrometers (precipitable) / meter squared. In other words, you take a square meter of Martian surface, and extend a column from it up to space. Assume all the water in this column of air precipitates down onto the surface. 1 gram of water would form a micrometer thick layer across this square meter.
The moisture in the Martian atmosphere is typically from 5 to 10 um pr / m^2.
References
"The Atmosphere and Climate of Mars", Edited by Robert M. Haberle, et all, Cambridge Planetary Press, 978-1-107-01618-7.