As a hydraulics layman thinking about hydraulic systems, it seems that the important factor is to have a liquid that doesn't compress much or at all. Doesn't water meet this requirement, and what other properties should the liquid have (if any) that water doesn't?
Water meets the low compressibility requirement, but there are many other considerations in the design of a hydraulic system:
Boiling point/vapor pressure: If the system warms up during operation, the fluid may boil, which results in high compressibility and thus decreased effectiveness of the hydraulic system. Hydraulic fluid has a higher boiling point than water to help combat this. Related to this is the concept of vapor pressure. Hydraulic systems often involve small orifices, which can cause cavitation (localized boiling). This cavitation has the same effects as boiling and can cause pitting damage to the components near the cavitated region. Hydraulic fluid has a lower vapor pressure which helps here.
Freezing point: It would not be a good thing if your car's brake lines froze every time it got cold outside. Most hydraulic fluids have much lower freezing points to prevent this from happening under normal circumstances.
Oxidation/corrosion: Water, being an electrolyte, will cause rust inside the lines as soon as air inevitably leaks into the system or the system isn't bled properly. Water will also exacerbate galvanic corrosion when dissimilar metals are used in the system.
Lubrication: Hydraulic components use seals and often involve sliding interfaces (cylinders and spools, for examples). Using an oil as the fluid means the working fluid can also function as a lubricant.
Organic growth: If perfectly distilled water and a closed system could be guaranteed, this would be a non-issue. But in practice, this is never the case. Oil-based hydraulic fluids are much less conducive to organic growth than water.
Water is used in some systems where other considerations trump these (for example, some food-grade applications), but for a wide variety of applications, oil-based hydraulic fluids are the better choice because of the design considerations above.
In addition to the other answers, water is commonly used as a primary working fluid in industrial hydraulic systems with additives to prevent the issues mentioned above. Typical applications include furnaces, foundries, plastic extrusion, etc. Anywhere that standard hydraulic oil represents a major fire or contamination hazard, yet operating costs do not justify purchasing synthetic fluids for $1100+/barrel.
Factory hydraulics are ideal for water-based fluids because the environment is controlled. Atmospheric temperature is typically maintained within 15C-30C year-round. So boiling and freezing are non-issues. Oil properties can be continuously monitored for changes in pH, composition, or particulate levels. New water can be regularly added to the system to replace losses from evaporation.
The percentage of water in a typical mixture may be between 30 - 80%. As the fraction of water increases, the viscosity of the mixture decreases. Low viscosity is the primary challenge of water-based hydraulics. Modern hydraulic pumps (axial-piston, radial, vane, etc) are designed with internal leakage passages for cooling. Somewhere between 2-6% of total flow is "leaked" or intentionally siphoned away from the flow outlet for cooling of the pump's sliding surfaces. The flow rate through these passages is obviously dependent on viscosity. The problem with running 100% water is that viscosity falls below 1.0 cSt at 20C. Compare that to 46 cSt viscosity of standard hydraulic oil. The machining tolerances of pump surfaces made for pure water would be extremely fine (<1 micron), or leakage could siphon away the majority of total flow and surfaces could frequently experience contact due to the thin film of water separating them.
Current manufacturing tolerances in series production are simply not that good yet. There are specialized pump and valve manufacturers like Oilgear and Dynex-Rivett with more expensive product lines targeted at the extreme end of the water-based fluid market (80-95% water). There are also products like the Danfoss PAH pumps which can operate at reduced pressures with pure tap water. However a PAH costs approx 25x more than an equivalent-displacement pump for standard fluids.
Source - my former boss spent years trying to redesign standard axial-piston pumps for 100% water
In terms of historical interest, water was used in the first hydraulic brakes and very soon the obvious problems of freezing (due to ambient temperature) and boiling (due to lots of brake use) became evident.
To begin avoiding freezing, alcohols and glycerol/water mixtures were used, but issues were still evident.
Leaking was one, but seals were soon developed to cure that, back in 1921.
This led to the progression to the current options of mineral oils (Citroen still uses these) as well as glycol/ether and silicone products.
Even the presence of water in hydraulic oil will damage the system components. Then you can think what will happen when you choose water as a hydraulic fluid. A hydraulic fluid is selected depending on the application the system will perform. For systems operating under high temperature, fire resistant fluids can be used and for systems operating under low temperature petroleum based fluids can be used. Water is considered as a wrong selection because of many reasons. Hydraulic fluid has many functions like providing lubrication and sealing, heat transfer, and power generation. But, while using water most of these functions can’t be achieved. The water will reduce the strength of lubricating film and hence internal or external leakage will occur. Also, the water molecules will result in oxidation/corrosion of metal surfaces. The low boiling point and freezing point of water is another drawback of using water as a hydraulic fluid.