The rush in short-rush refers to hurrying city production, which can be either pop-rush or gold-rush. The short in short-rush refers to not hurrying the production all the way. Instead of rushing all the shields necessary for the item, you rush part of the shields, then let the project complete on its own.
The way that you typically do short-rushing, is by temporarily changing the production to something cheaper first. This cheaper item has the shield cost that you want to short-rush to. You rush the cheaper item, then immediately switch production back to what you actually want to produce.
There are two major reasons why you want to do short-rush. One of them is to get around the pop-rushing rule that you cant kill more than half of your citizens at once. For example, if you have a size 3 city, and you want to rush a 40 shield library, it would normally take two citizens to rush 40 shields, and youre not allowed to do it in a size 3 city. So, what you do is: let the city accumulate 1 shield first, then switch production to a 20 shield item, like a spearman, and rush it. This will cost 1 live, and bring the city size down to 2. Then on the same turn, switch production to library. Because you already have 20 shields in the box, it takes only 20 more shields, or 1 more live to rush it. With a size 2 city, youre allowed to do that.
@Pentium: youre thinking of early patches where pop-rush gives 40 shields for the first live. Thats not the case anymore.
The 2nd reason to short-rush is to save gold during rushing. Lets take the example of a city that makes 20 shields-per-turn, and is building a cav. After one turn that accumulated 20 shields, you want to rush the cav to completion. If you rushed all 60 remaining shields, then you end up wasting 20 shields of production on the next turn. So what you should do is short-rush to a 60 shield item, like a musket, then switch back to the cav. On the next turn, the city will generate the remaining 20 shields necessary for the cav, and completing it. Bingo! Short-rushing just saved you 80 gold.
The 3rd reason to short-rush is to save shield over-runs. Lets take another example. This time we have a city making 7 shields, and were building a 30 shield swordsman. The swordsman normally would take 5 turns to complete, and waste 5 shield due to over-runs. To save time and save shields, we short rush to 10 shields (using the 10 shield worker) after 1 turn. This way, we have 20 shields left to go. At 7 shields per turn, it will only take 3 more turns, and wasting 1 shield at the end. By short rushing, at the cost of 12 gold, we produced the sword 1 turn sooner and saved 4 shields.
edit:

huge cross post, you guys type fast.