To the casual traveller, a wormhole is just a tunnel between two patches of space. Both sides of the tunnel look the same.
However, like a ski resort full of girls hunting for husbands, and husbands hunting for girls, the situation is not as symmetrical as it seems.
During their formation, wormholes are directional. One side is the "entrance" and the other is termed the "exit". When they are fully formed these two ends behave the same for all practical purposes, but we keep the distinction in their labeling. The exit is always labeled "K162". The entrance can have many names, for example "B274", for our highsec static in Seclusion. You find this in the info panel for the wormhole:
The entrance label allows you to identify the type of wormhole, and get a lot of information on it.
So, how do wormholes form? Well, first the entrance appears as a cosmic signature on your probe scan window. At that point, the K162 exit has not yet formed - there is no exit hole yet. There is only the embryonic entrance hole.
Our warp engines perturb the embryonic wormhole. If you warp to the cosmic signature then even if you do not enter, a spacetime link is created to the destination system. However, the wormhole exit is not visible yet in that system. As the link ages, there is a certain probability per unit time for it to become visible, even if you leave it alone.
Now, if you enter the nascent wormhole, and you are the first one to go through the wormhole, then at that particular moment, the exit hole will spawn visibly in the destination system. Folks there see a new signature pop up on their probe scan window at that point. That is why wormholers get really nervous when a new signature appears. It could be something that appears spontaneously - or it could be someone from another system who just breached a new wormhole and landed in your system.
I have found a diagram to explain wormhole spawn mechanics, on the galnet forums, and I am including it here for completeness' sake:
