Short channel effects arise when the channel length (the distance between the source and drain of a mosfet), is of a similar order of magnitude to the depletion layer (the area between a p type and n type material that has no charge carriers). This causes it to behave differently to long channel devices. One of the main differences is that short channel devices have an early saturation current.
In a long channel device, the saturation current has a quadratic dependence with the gate-source voltage. What does this mean? Well, for a given gate-source voltage the drain-source current will increase with drain-source voltage, but only up to a certain point. After a certain point the drain-source current will stop increasing as you increase drain-source voltage, this is because the inversion channel connecting the drain and source "pinches off" due to the electric field between the gate and source. You could assume that at this point the drain-source current should be zero, but When the electrons reach the pinch oﬀ point they are injected in the depleted region and travels to the drain. Anyway, as you increase the gate-source voltage, the point at which the saturation occurs is at a higher current, and the relationship between the saturation current and the gate-source voltage is quadratic.
In short channel devices the saturation current has a linear dependence with gate-source voltage.
This is because the current saturates prematurely mainly due to velocity saturation. This is happens because the velocity of the carriers (and therefore the current) tends to saturate due to scattering effects (collisions suffered by the carriers).
Another short channel effect is that there is a lower drain-source voltage needed for "punch through". This is where the mosfet will start conducting even if there is no gate-source voltage. This happens because the depletion region at the p-n junctions of the source and body, and drain and body both become larger as the drain-source voltage increases. At a certain drain-source voltage these depletion regions will merge together and any electron injected into this region can travel through the mosfet. This means that when this happens the current through the mosfet becomes independent of the gate-source voltage. For shorter channel devices the depletion regions will merge together sooner because they are already closer due to the fact that the drain and source are closer together.
I believe drain induced barrier lowering is the name of the effect that causes this lower "punch through" voltage.