Before we discuss counter builds, let's talk about what an early Zealot rush is:
"my defenders got wiped by 4-5 zealots and more and more zealots kept pouring in and eventually killed me"
This is the dreaded 2 Gate rush.
Your first goal is going to be identifying it. Luckily this is fairly easy to do. If you scout with your 12-13th drone/probe/scv you should get to your opponents base while his first Gateway is still building, worst case while his second is. All 2 Gate rushes start the same: Pylon, Gateway, Gateway, Pylon. This makes them very easy to pin down. If you see that second Gateway go up before the Cybernetics Core you know he's playing some sort of rush (pro tip: you can click on a Protoss building warping in to see what it will be). If he does a delayed rush, chances are you'll already be in a better position to defend, so you really want to watch out for this early rush.
The attack that will come should be 5 Zealots with 2 more in route (obviously he may wait for more or less, but this is the best time for him to attack).
As Zerg a 2 Gate opening is one of the scariest things to deal with (right after Hellion harass into Mech). If you fast expanded, use the expo as a damage soak and cancel it before it gets killed (if you can). Ideally you'll want to keep that expansion alive as his economy will already be devastated. When you scout a 2 Gate, you'll want to build a Spine Crawler almost immediately. Additionally, a Roach Warren is also recommended. Roaches are very strong vs Zealots (slow attack and starting armor vs a unit with fast attack) and if you can position some by your ramp or other choke you can cut the surface area of your Roaches in half. If you're already into a more speedling based build, then you're going to want to wait for him to surround your Hatchery or engage your Spine Crawler. This will naturally give you a very nice surround on his units as they will spread out. I also recommend bringing your Queen along for ranged dps (and to infuse the Spine Crawler).
The key thing for Zerg about repelling a 2 Gate play is the recovery. Early 2 Gate is so effective that even though he is putting himself at an economic disadvantage, he may do enough damage to your economy to win. You'll want to do the following: Put an Zergling outside his based (this is a scout for his next move); Tech to Roach or Hydra or Muta, as 2 Gate will likely lead into 3 Gate, 3 Gate + Robo or 4 Gate; Power drones, this one is key as he's making a huge economic sacrifice to push with 2 Gate and you need to exploit this for victory.
Terran traditionally have less problems with 2 Gate as they can usually wall off before the Zealots get there. If you already have Vespean Gas then its ok to go for Marauders with Concussive Shell, however, if your strategy didn't include that, you're going to want to skip add-ons and throw down a second Barracks when you scout the second Gateway. With Marines faster build time, you can create a tough Marine force behind your wall well before his Zealots arrive. This ruins the fact that he made an early sacrifice to go for a Zealot push and should give you command of the match. From here is very easy to transition into an MMM or even BioMech which should give you a quick victory.
As Protoss you're going to also want to wall, stick a Zealot or two in the opening and push for Stalkers. If you can have enough 1 Stalker out and your opening plugged before he gets there, chances are you've already won. It is not unusual, upon seeing a 2 Gate, to throw down a second gateway yourself. As defender you'll naturally have the advantage as his troops have a longer rally distance, and you can control the choke. If you push out an early Sentry, he'll probably Rage Quit and call you a cheater.
Now if you're dealing with a Proxy Gate (which is different than a 2 Gate) you have slightly different things to watch for. Lack of Assimilators and Pylons in his base are your best clue. At this point you'll want to start taking your scout and covering all the corners of the map. Additionally move a scout to outside your natural or the nearest Xel Naga tower.
For protoss you have to make excellent use of Sentries to delay your enemy. If possible, try to split his army in two with Force Field(s) and take on that half by itself with your zealots, sentries, and stalkers. You won't be able to take him in a head-on fight as Marines/Marauders dominate your T1 units.
Use this delaying tactic to get a few Immortals as soon as possible and transition into Colossus with the range upgrade. The splash damage is very effective against the tightly clumped Bio Ball, while Immortals can take many hits from Siege Tanks. Use Blink Stalkers to get close to the tanks and focus them down. Note that once you have a few Colossi, your opponent will usually get a few vikings to counter them, but focus them down quickly with your stalkers.
Honestly, it is a very difficult thing to beat as Protoss if your opponent is of better or equal skill, especially since you have to tech to very expensive units just to counter his (mostly) T1 army. You'll just have to delay and outplay them and not make any mistakes, and make sure you are very proficient with your Force Field and Guardian Shield usage.
Another option is to go High Templars with Psionic Storm and Feedback on the Medivacs, but I rarely go this route as it is too gas-intensive and higher level Terran players will just use Ghosts which nullify High Templars with their EMP ability (Removes all energy and 100 shield points). Not to mention if your clump of High Templars goes within range of a Siege Tank you're dead meat.
Best Answer
As mentioned in the Reaper question. You need a stalker to deal with early Reapers as Photon Cannons tend to set you back by too much. However, there is no reason it has to be the first unit you build.
Since a Stalker requires a Cybernetics Core (60 second build time) and it in turn requires a Gateway (which can build Zealots with a 33 second build time) you can easily build a Zealot while waiting for the Cybernetics Core to finish.
So the better question is: "Why wouldn't you?"
Obviously if you can't afford another Pylon and only have 2 Psi to spare it makes sense to cut the Zealot. But consider also the costs to build a Stalker: A Vespean Assimilator (75 minerals), Probes harvesting Gas instead of Minerals, the additional cost of the Cybernetics Core (150 Minerals). These all add up. That said you can (if you play decently) afford both.
Other things to consider. Once your stalker is built what are you going to build next? "Standard" play is usually to add on another Gateway (150 Mineral cost). By cutting that Zealot you can start building a second Gateway before your Stalker is ready. You'll probably also need more Psi at this point, and a Pylon has the same cost as a Zealot. Its worth noting that you should have a better economy once the Stalker is finished (you were building probes this whole time right?), so these things will still be affordable.
So to answer your unasked question: Should you cut the production of the first Zealot?
Try it out! Play a couple times where you build both, and where you only build the Stalker. How much Minerals do you have? Does it feel like enough for the rest of your game plan? Would 100 Minerals help you? Would you rather have that Zealot to help block your entrance?
This all depends on what your strategy is, so what you need to do is try both and decide. They're two options and there is no correct answer.
For reference material here is Tester vs IntoTheRainbow in the Razor King of the Beta Tournament.