Standard normal intuitive understanding What does it mean for a standard normal to have mean 0 and standard deviation 1? 
I'm having trouble understanding - what is a "normal variable"?
 A: "Standard normal" is the name for the normal distribution with parameters fixed at values $\mu=0$ for mean and $\sigma=1$ for standard deviation. This simplifies the probability density function to
$$
\phi(x) = \frac 1{\sqrt{2\pi}}e^{- \frac 12 x^2}
$$
By convention, we usually use $\phi$ to denote the standard normal probability density function and $\Phi$ to denote standard normal cumulative distribution function.
A: A "normal variable" or, more precisely, a normally distributed variable, is one that follows a Normal distribution. The Normal distribution is a particular kind of bell-shaped curve (the formula is given by @Tim in his answer). Many natural phenomena follow a Normal distribution (at least approximately).
A standard Normal is a Normal distribution that has been standardized to have its mean = 0 and its standard deviation = 1; any Normal distribution can be turned into a standard Normal by subtracting the mean and dividing by the standard deviation. 
A: Let's say we are both part of a group of 100 people whose weight can be assumed to follow a normal distribution with mean 60 kg and standard deviation of 5kg. In other words, the average weight for this group is 60kg but the individual weights can be as low as 60kg - 3x5kg = 45kg and as high as 60kg + 3x5kg = 75kg. (Most observations are expected to fall between "mean - 3 standard deviations" and "mean + 3 standard deviations" for a normal distribution.)
If your weight is 50kg and my weight is 70kg, it is clear that your weight is below the group average and mine is above the group average. But how far below the group average is your weight and how far above the group average is my weight? We can use the standard deviation as the yardstick:
(50kg - 60kg)/5kg = -10kg/5kg = -2 
(70kg - 60kg)/5kg = 10kg/5kg = 2 
So your weight falls 2 standard deviations below the group average and mine falls 2 standard deviations above the group average. We can say that our standardized weights are -2 and 2, respectively. While our original weights were expressed in kg, the standardized weights are unitless.
If we standardize every single one of the weights of the 100 people in the group following the same process as above, we'll obtain a set of 100 numbers expected to range between -3 to 3. The average of these numbers will be zero and their standard deviation will be 1. In other words, the average standardized weight of the group will be 0 and the standardized deviation of the standardized weights will be 1. The distribution of the 100 standardized weights will therefore be a standard normal distribution.  
The standard normal distribution arises when we standardize observations (be them weights, heights, etc.) which are known to follow a normal distribution with mean mu and standard deviation sigma. The standardization is accomplished via the formula (observation - mu)/sigma.  The standardized observation is known as a z-score and tells us how far above or below the mean mu the original observation is as measured against the yardstick of the standard deviation sigma. If the z-score is strictly negative, the observation is below the mean mu. If the z-score is strictly positive, the observation is above the mean mu. If the z-score is zero, the observation coincides with the mean mu. 
