A short is a direct connection between any 2 points in a circuit. Often it is used to describe something unexpected, or in error, but it can be used to describe an expected situation (as in "You expect to see a short circuit between the ends of a power cable", or "if the fuse doesn't look like a short circuit, it's blown").
A short (which is an abbreviation of "short circuit") is not necessarily going to damage anything, but it represents a low (often near zero) resstance, so where there is a potential difference between the points shorted, and where the source of that potential difference can supply significant current, bad things can happen.
A power supply is an example of a device which supplies a voltage (the potential difference) and which can supply possibly significant amounts of power. Some power supplies cannot tolerate a short and the excessive current drawn will damage them. When the power supply is your expensive (well relative to a wall wart) then you don't want to risk it.
A wall wart may not be designed to handle overload conditions either, and it may be damaged by excessive current draw (a short being the most extreme example). But at least it's cheap.
The real solution is to have a voltage regulated and current limited (perhaps foldback current limited) power supply. These are typically built to handle short term overloads.
The power supply surviving is a good thing, but the short can damage components on your circuit under development on a breadboard -- but that's the risk you take. With leaded components, it is very easy to accidentally short (say) one resistor lead to another in the rats nest that breadboards can become.
How do you know if a wall wart is regulated? Easy! Measure the voltage on it without any other load. If it is very close (say under half a volt) to the voltage stamped on it, then you can be pretty sure it's regulated.
If the wall wart is very light, then chances are that it is a switchmode regulator and it will be regulated (another dead give-away is an extended input voltage range -- say 90 to 250 volts).
Heavy wall warts contain a mains transformer (these are becoming rarer and rarer) and will often have a voltage reading under no-load conditions of between 20 and 50 percent higher than the rated voltage (so a 12 volt wall wart could measure between 14.5 and 18 volts -- probably closer to the latter).