If water is not a good conductor, why are we advised to avoid water near electricity (no wet hands near circuits etc.)? How can water be a medium to conduct current while its ionisation is so negligible that, in principle, no current should flow?
 A: "Pure" water is a very poor conductor (resistivity is actually used as a measure of purity).
"Real" water is not pure - it contains electrolytes and is quite conductive. Also - when your skin is wet, its resistivity is significantly lower.
For example - "pure" water has a resistivity of (about) 18.2 M$\Omega\cdot\rm{cm}$. With 10 ppm of dissolved NaCl ("very good quality tap water" would have less than 50 ppm), the resistivity drops to about $43~\rm{k\Omega\cdot cm}$
A lot of detail can be found in "Conduction of Electrical Current to and Through the Human Body: A Review" (Fish and Geddes, ePlasty 2009, 9, e44). Table 3 sums it up:

Table 3
Why immersion in water can be fatal with very low voltages


*

*Immersion wets the skin very effectively and great lowers skin resistance per unit area


*Contact area is a large percentage of the entire body surface area


*Electric current may also enter the body through mucous membranes, such as the mouth and throat


*The human body is very sensitive to electricity. Very small amounts of current can cause loss of ability to swim, respiratory arrest and cardiac arrest

Table image
A: Water is indeed NOT a good conductor UNLESS impurities exist. And if you could keep the impurities from contaminating it, you could put a radio in it and it would still work. You can find out yourself. Take distilled water (about a cup) and check the resistance with an ohm meter with probes fixed distance approximately 1/2 inch apart. It will show infinite resistance for all practical senses.
Then dissolve about 2 grams of salt in a tablespoon of water. Keep the probes in the water and add the saline to the water and watch. The ohms dive from being an insulator to a fairly good conductor... You can even do this by just putting your hand in the distilled water as the salts from your skin dissolve in the water. 
The argument that water is a conductor because "water seldom is pure enough not to contain some dissolved mineral contaminants" is like saying that it isn't clear either because look at the oceans or the lakes. 
A: Water is a poor conductor compared to wires and metal, but is an excellent conductor compared to air, glass, ceramic tile, and other objects in the bathroom or kitchen, particularly as mentioned in other answers if there are certain impurities dissolved in it.
When our bodies and feet are wet and we are walking across a wet floor, and we have somehow touched a wet electrical outlet or wire attachment; if the outlet is not ground fault protected, the electricity from the hot (live) side of the outlet will try to find a shortcut to ground through your wet skin and your wet feet to the floor to ground.  As mentioned earlier, even a very small current through your body can stop your heart (cardiac arrest).
That is why GFI (ground fault interrupter)  protection is required building code on all electrical outlets in kitchens and bathrooms.  This type of protection senses when there is a short circuit bypass from the breaker ground and cancels the hot line on the outlet.
A: The problem is in the definition what a "good" or "bad" conductor is.
In school children are taught that everything is quite easy: there are "good" conductors like metals and "bad" conductors like plastic.
Short question before: What do you think is the difference in electrical conductivity between silver (the best non-exotic conductor under standard conditions) and stainless steel ?

 Silver has a conductivity of 63 000 000 S/m, stainless steel has 1 450 000 S/m, silver conducts electricity 43 times better than stainless steel! (I am writing the numbers out because the difference is hidden for laymen if you use unfamiliar scientific notation).

S is an unit called siemens, and siemens per meter (S/m) is the SI unit of conductivity.
The thing is that electric conductivity is one of the physical quantities which has the largest observable differences between various materials. I give a short overview:
Semiconductors (They are still looking like metals!)

*

*Germanium:  30 000 000 ($3\times10^7$) times worse than silver ($2.17$ S/m)

*Silicon:    40 000 000 000 ($4\times10^{10}$) times worse than silver ($1.56\times10^{-3}$ S/m)

Bad conductors

*

*Seawater       13 000 000 ($1.3\times10^{7}$) times worse than silver ($4.80$ S/m)

*Damp wood:     100 000 000 000 ($1\times10^{11}$) times worse than silver  (From $10^{-4}$ to $10^{-3}$ S/m)

*Glass/rubber:  ~100 000 000 000 000 000 000 ($1\times10^{20}$) times worse than silver (From $10^{-15}$ to $10^{-11}$ S/m)

*Air:           ~1 200 000 000 000 000 000 000 ($1.2\times10^{21}$) times worse than silver ($3-8\times 10^{-15}$ S/m)

No, it is not a joke. Let's imagine that the distance from you and the moon is comparable to a good conductor. The school definition of bad conductor would be then the height of an apple tree for sea water and the size of an atom for glass.
So why are we not aware about this staggering amount of difference? The reason is that we are quite fragile about electricity, even very small currents could cause pain or death. So water has still enough conductivity to cause problems despite being a comparatively bad conductor, metals are simply extremely effective conductors allowing to transmit electricity over vast distances with relatively small losses. The other materials as non-conductors like glass or air are effectively non-conductors even shielding someone from very high voltages (The problem is once a connection has been built, an arc of ionized molecules is forming which has much, much better conductivity than the unchanged material. Normally you need approx. 5 000 V per centimeter in air to build a connection so you would be safe from 1 000 - 10 000 V of transformer stations, so this is the reason they put up the danger signs).
A: In high school my chemistry teacher, Mr Stratton, set up an experiment.  He had a light bulb fixture on a wooden paddle, with a power cord attached and two metal prongs projecting at right angles, such that the paddle could be set over a beaker of water with the prongs dangling into the water.
He first filled the beaker with distilled water and screwed a 40w incandescent bulb into the fixture -- no light.  Then he tried something like a 10w bulb -- still no light.  Then he tried a neon light -- very dim glow.
Next he poured in a small amount of salt (NaCl) and swished it around -- the neon light began to glow brightly.  Tried the 10w bulb -- glowed at nearly full brightness.  Tried the 40w bulb -- not fully bright but at least halfway bright.
Water is a good polar solvent, and combine that with any sort of chemical which ionizes when dissolved and you have a solution which can carry a significant current with relatively low resistance.  (And salt-like chemicals are everywhere.)
A: Water is a conductor! Because water seldom is pure enough not to contain some dissolved mineral contaminants (salts / electrolytes). Standing water in a lake or pond or puddle is not almost never pure but never pure. Dirty water qualifies as impure water, by the way. Understand then that the contaminant is in the form of ions which make water a perfect conductor as it has another great quality: adhesion to solid or supple surfaces (human skin for example). Hence creating the superb if not the perfect condition of making water an ideal conductor of electricity to shock your backside into oblivion when there are raw high voltage wires from downed transmission lines/poles due to a storm or random car accident in or near a body of water as in puddle in the street.
