Overview
As with any same species encounters, Protoss vs. Protoss battles are more a race for resources and unit production than anything else. The capabilities at hand are the same for both sides so the player that knows his/her force the best, both their strengths and weaknesses, will usually be the victor.

Zealot Rush
In most Protoss on Protoss encounters, the first few minutes of the game will usually consist of Zealot Rushing (at least on land maps). These encounters will typically boil down to numbers or micromanagement.

If you find yourself outnumbered when rushing into another player's base, your best option is to go straight for your opponent's Probes and slow down the resource production as much as possible. Make sure you command your Zealots to attack the Probes directly and not with an Attack-Move command to ensure the fastest and most efficient kill rate. Try and avoid direct Zealot engagements in this type of situation since not only are you outnumbered, but you will be unable to reinforce your troops in time, while your enemy will have their Zealots producing in close proximity. If you find yourself outnumbered in your own base, try and initiate Zealot on Zealot encounters using Hit and Run tactics until you've produced enough Zealots at your own Gateways to counter the invading force. Try to avoid sending your Probes to attack the Zealots unless it's a last resort or if the enemy Zealots are heavily damaged since doing so would be like sending lambs to the slaughter.

If you outnumber the enemy when invading, kill the enemy Zealots as soon as possible then go straight for the mining operation. Reinforce your troops as quickly as possible to make sure the pressure on your enemy doesn't let up and in most situations, you'll be able to finish them off quickly. Watch for escaping Probes, but don't send all of your Zealots chasing after them. It's better to finish off the enemy Nexus first to ensure they can't return to it later on, and even if they have the resources to build another one, it will take some time to Warp it in from Aiur, time you can use to find and hunt down the wandering Probe(s) after you've finished off their original base. If you outnumber the enemy on defense in your own base, engage the invasion force as quickly as possible and do all you can to prevent them from reaching your mining Probes.

Zealots and Dragoons
If you make it past the early game Zealot rushing, you'll want to begin supporting your Zealots with Dragoons. They're useful for getting free hits in during Zealot on Zealot encounters and in numbers with the range upgrade, they can heavily damage Zealot forces before they can even reach the Dragoons. Also if your Dragoons survive into the late game, they serve as excellent anti-air units and are particularly useful for fending off the Reaver Drop. In the latter game, you may want to concentrate production more on Dragoons than Zealots, but the combination of the two is undeniable throughout the entire game, so don't neglect to make a few Zealots even in the late game.

Detection
With the Dark Templar rejoining their Protoss brethren on the battlefronts, detection is an important issue in the early-mid game. The need for Photon Cannons is essential for base defense and you'll need them in numbers. If possible, keep a small cluster of Photon Cannons (since a single Cannon is almost useless against even a single Dark Templar) close to your mining operation and watch and listen for signs of Dark Templar attack. Other than the Dark Templar, the only units with cloak abilities are those that are within an Arbiter's field. You can combat that method of cloaking either by chasing away or destroying the enemy Arbiter or have an Observer/Photon Cannons nearby, but with the cost of Arbiters as well as their production position in the tech tree, you probably won't encounter them in the majority of your battles.

ScoutCraft
Most of the Protoss ground units have difficulty handling aerial units (except maybe for the Dragoon and Archon) and the Scout can be especially useful in taking advantage of that problem. Not only can they attack both ground and air targets, but their speed and manuverability make them the perfect unit for scouting around the map and quickly shutting down enemy expansion attempts (which is extremely important considering the speed at which a single Probe can set up a stalwart defense and mining operation). Their speed also comes in handy for quick retreats to Shield Batteries and moving back to the forefront of battle. Once gathered in numbers, Scouts can be deadly against any force, and the more upgraded they are, the more destruction they can cause. Groups of 18 or more Scouts are extremely difficult to eliminate, even with the mighty Carriers, but be sure to keep them together for maximum efficiency.

Reavers
The Reaver is a powerful weapon against any enemy Protoss force. Use the Reaver Drop to quickly destroy mining Probes at enemy bases. They're also very effective for quickly destroying enemy Shield Batteries and Photon Cannons. Tightly packed enemy ground units are also prime targets for Reavers. However, with their slow movement, be very wary of enemy Psionic Storms as well as Scouts and Carriers.

Arbiters
If you do get the opportunity to produce Arbiters, you can gain certain advantages in battle. If you can clear out an enemy's detectors, by destroying their Observers and/or Photon Cannons, units beneath the Arbiter's cloak field will be able to quickly destroy an enemy force or base. However, an even better use of the Arbiter lies in the ability to bypass enemy defenses with an Offensive Recall. This will allow you to quickly transport troops right into the middle of an enemy base, or across the map in an instant and without having to risk losing units inside Shuttles making a run past a Photon Cannon defense. The Arbiter's Stasis Field ability is also very useful against groups of Carriers, giving you enough time to set up the proper countermeasures.

Psionic Storm
The High Templar's Psionic Storm ability is probably the most effective tool to use later in the game against other Protoss. It causes massive damage to every Protoss unit, especially if they're packed together. Casting Psionic Storm over a busy mining operation can reduce the enemy colony's output in half (if not more) by killing all the Probes beneath it. It's also one of the better counters against a Reaver Drop (especially if you can manage to hit both the Shuttle escort and its Reavers in one cast. Also against large fleets of Carriers, Psionic Storm can help save your base by heavily damaging all Carriers within the area (which is fairly easy since the Carriers tend to "stack" on top of each other) as well as the attacking Interceptors. Psionic Storm is also key when defending your base, since Psionic Storm has no effect on buildings. Make generous use of it should you get invaded by Zealots, Dragoons, Carriers, etc.

Hallucination
The High Templar's Hallucination ability is also an effective tool to use against other Protoss. Use it with Shuttles to give them a better chance of surviving when running past a blockade of Photon Cannons. Hallucination also works well with Carriers, but usually with about a 1:1 ratio between Hallucinated and real Carriers, since 8 Hallucinated Carriers escorting only 1 real Carrier can look funny when only the 1 real Carrier begins deploying Interceptors. You can also use Hallucination to "enlarge" an Archon force giving your real Archons more time to dish out the damage.