Short-circuits are caused when unprotected wires touch each other, causing the current to 'jump' to another part of the circuit, without passing through all the intended components first.
Usually your short-circuits will mean current by-passing one or more LEDs - this is not a problem if it doesn't last long. The LEDs left in the circuit will be momentarily getting too much power, but this should not be a problem in the short-term. If you leave the short-circuit un-fixed, it will eventually cause the remaining LEDs to burn out.
The worst short circuit is one that causes your resistor to be bypassed. This should be avoided at all costs - it will usually kill your LEDs.
If you short-circuit your power supply (by causing + be connected directly to - ), you will quickly get a dead battery. You may also get sparks, heat and smoke.
TO BE AVOIDED.