The Zergling Run-by is a technique that consists of moving a group of Zerglings (usually six to twelve) past the Photon Cannons that protect a Protoss natural expansion, and sending them directly into the undefended main base in order to harass. This tactic is usually only performed early on in the game, when unit count is low.
One of the many early game counters to the Forge first Protoss build is to attempt a Zergling run-by. If done correctly, this can break the Protoss player and deal a large amount of damage, with very little risk to yourself. If the run-by doesn't work, you can minimize the damage that you do to your own economy. This is accomplished by powering Drones while attempting the run-by, and by pressuring the Protoss player into either building a Photon Cannon or pulling his Probes from mining.
This technique only applies at the beginning of the game against a Forge first build from the Protoss player. At any other point in the game, you should be trying to kill the Cannons, or otherwise "break" the Cannons, as opposed to running past them.
To last a parting vengeance and as a courtesy to your allies in team games, if you are facing imminent disaster and the opportunity presents itself, you can run your fast zerglings around the enemy units coming into your base to kill an enemies' Command Center, Nexus, or more rarely a hatchery. This hopefully will slow an enemy down enough to give your allies a better chance of winning the game.
If there is too much defence, or too many Cannons, you do not need to perform a run-by, however, you must decide how many Zerglings to produce before you can test the defence. This ability to threaten the Protoss without actually attacking allows you to keep your options open. If there is an opening to run by, the potential damage you can deal early on compensates for the Larvae that were used making Zerglings instead of Drones.
It is important to gauge how well the Protoss defence will fare against your Zerglings. Assuming there are no probes blocking, six Zerglings can easily run by or destroy one Cannon, and twelve can easily run by or destroy two. If probes are blocking, your ability to run by becomes highly dependent on micro, Sim-City, and map architecture.
If your Zergling run-by is successful, you should try to do as much damage as possible, as well as getting scouting information. Being able to kill stray Probes, and possibly snipe Tech buildings, Assimilator or Pylons if the opportunity presents itself.
If it was unsuccessful, then you should prepare for a counter attack. Hopefully, you didn't hurt your economy that much with your attempted run-by, and you can continue along your Tech path with adequate timings.