What are effective PvP strategies in Minecraft?

A few simple tactics:

If you are ambushing somebody, make sure to have a splash poison potion (I or II), as they ignore armour, which can be extraordinarily helpful against opponents with enchanted armour. It does not kill them, though, only brings them to half a heart! This tactic is also useful is you are turning round to face a pursuer.

Enchantments are also an excellent way to defend yourself and increase your attacking potential. A fire aspect enchantment on your sword is a great alternative to a poison potion, but it can be harder to find as it isn't a guarantee that you'll enchant it on the first go. Make sure to also keep up your defensive enchantments, though!

If you are currently in a melee battle, try and circle+jump around your opponent whilst spamming your left click. This is a common tactic though, so expect your opponent to do it as well! The jumping ensures you will get critical hits, which can get you up to 50% extra damage! Try and switch directions often though, as it is easy to counter by simply hitting a bit in front of where you actually are.

It can be a lot easier to kill them with a bow, which negates the danger of the enemy quickly wearing down your health, unless they are a better shot than you. You can shoot them a few times before entering melee combat, as they will enter with slightly diminished health. If you have a fire aspect enchantment on your bow, then they will enter slowly burning to death. It is also a great way to finish off a fleeing enemy.

If you're on the defensive, remember to block/parry! You simply hold right click facing an enemy, and the attacks only do half damage. This does not decrease durability on the sword.

There are a few good tactics to PvPing,

Strafing: Do quick circles around your enemy while focusing and hitting the target at the same time.

Pots:Bring strength 2 for every fight to get an advantage, bring speed 2 for an edge on your enemy, or a quick get-away.

Weapons/Armour: Do not put any enchantments on close range weapons other than sharpness, other enchantments will weaken the power of your sharpness.Bows do not have the same qualities as swords and axes, so go all out on the enchants. Go for protection 4, unbreaking 3 armour. You could use thorns to cause your opponents armour slight damage with every hit they take, however with protection 4, unbreaking 3 and thorns you can't repair your armour in an anvil.

Crits/jumping: Do not use this tactic ever, you take more knockback in pvp (you don't want to be knocked back in pvp) and you do more damage, but your opponent also does more damage to you, so it's counter effective.

Don't get Shot: If you are in pvp with a good archer, you need to dodge those arrows. Run in a zig-zag pattern to throw off their aim.

Don't let your armour break: If you can tell that your armour is weak, Run! The worst thing that could happen in pvp is losing a helmet or boots.

Mcmmo: If you play on a server with the Mcmmo plugin, be sure to level up your skills in this order. 1.Unarmed 2.axes 3.archery(only if you're good at it) 4.Acrobatics.

Identify Hackers: wondering why you are getting hit while a player isn't looking at you? Well those are hacks, and they are very common in pvp, eating/drinking and hitting at the same time is not possible withought a hacked client. Somebody head is twitching and snapping from player to player? That's a hack. Don't use hacks yourself.

Gapples: if you don't know what a gapple is, it's an enchanted golden apple. If you have them, use them. Don't waste gapples and exhaust your resources. If you have a decent gold farm don't be shy to use your gapples.

Hope these tips will help you ;)

A lot has changed since the other answers were valid. In particular, the Combat Update.

Shield is important. Spam-clicking is the most ineffectual way to deal damage - observe the tiny bar over your cursor as it fills up, it's percentage of damage you'll deal to your target, so time your attacks right. Potions are nice but far from game-changer, plus they don't stack and take a long time to drink, but splash potions can help in a pinch; regeneration will be more helpful than instant health, as it will heal more. The bow is still helpful - more so than trident or crossbow, if fully enchanted.

Then there's a whole range of additional techniques.

Probably the most important is placing blocks. By placing blocks strategically under you when jumping you may break your pursuit. You can destroy enemy's lava source (oh, yeah, a bucket of lava is a MIGHTY weapon against an ill-prepared opponent!), you can hide from ranged attack and get a bit of time to recuperate health.

A water bucket will put out fire, disrupt the opponent's attack, destroy their lava, and put out your fire.

Good food - golden apples, golden carrots are best for combat, though steaks or porkchops are fine too. And Enchanted golden apples - usually the player with significantly more enchanted golden apples is the one who wins.

With both combatants fully stocked up, and with a good stack of golden apples each, the one who loses the armor (due to durability damage) will lose, so if you're playing on that level, a spare set or two might be needed, though in usual PVP just a good, well-enchanted set will suffice.

Chorus fruit can get you out of a dead end or if the opponent managed to box you in with slow to break blocks. Enderpearls also help escaping a precarious situation.

And then there's one really mighty weapon: End crystals. The preparation takes some effort - either naked bedrock, or obsidian blocks, but if you can get the enemy there - the explosion effect is applied to player's feet, so if you hide your legs behind a bedrock or obsidian block, you're protected from most of the blast. Place and punch an End crystal, you'll deal massive damage, and there's almost no cooldown between the actions so you can spam them really fast whittling the enemy's health down. If fighting in the Nether or the End, you can use beds for similar purpose, although they don't stack, so they aren't nearly as convenient.

And of course the best way to win at PvP is not to fight at all. If you have an opportunity to build a deadly trap and bait your opponent inside, that's great! Though trap-building is a subject for a different post.