I'm not a judge, but I am often the rules guy at my tables. One of the most common questions I get is the difference between Activated abilities vs. Triggered abilities. Let's get in to this, and while we're here, let's cover the other ability types as well!

Activated Abilities

Comprehensive Rules 113.3b: "Activated abilities have a cost and an effect." You have to do something, like pay mana, and then an effect occurs. An activated ability will always have, and can be identified by, a colon (":") in the test. The simplest example is probably something like Crypt Ripper.

Simple. Pay , and Crypt Ripper gets +1/+1 until end of turn. Very often, activated abilities include a tap:

Note Arcanis has two activated abilities. The first is "tap Arcanis: draw three cards" and the second is "pay : bounce Arcanis back to hand".

The comprehensive rules really get into this in rule 602, but a few more things.

Most of the time the cost is in front of the colon and the effect is after. However, sometimes there's text after the colon which clarifies something, like when an ability can be activated or who can activate it (normally only the card's controller can activate an Activated ability).

Activated Abilities use the stack. When you pay the cost (and you must, before putting the ability on the stack), it goes on to the stack where it remains until it resolves, is countered, or otherwise leaves the stack. Also, note you can only activate an Activated ability when you have priority, but unless otherwise stated, they work at Instant speed.

Finally, you must be able to fully pay the cost in order to activate the ability. An Activated Ability which requires tapping a Creature follows all rules (it can't do it if it's got summoning sickness unless it has Haste,) and one which requires you pay two life means you must have two life to pay - if you've only got one life, you can't do it (even if you have Plantium Angel on the Battlefield!).

Triggered Abilities

Comprehensive rules 133.3c: "Triggered abilities have a trigger condition and an effect." They can always be identified by the use of one of three words: "when", "whenever", and "at".

(Algae Gharial was nuts in Shards of Alara limited.) Pretty simple. Another Creature dies, the Croc gets a +1/+1 counter. Triggered Abilities use the stack, and when the conditions are met to trigger the effect (another Creature dies) the ability goes on the stack and resolves (or is countered or whatever) as normal.

Probably the most common Triggered Ability is an Enters-the-Battlefield ability.

When the Beast enters the Battlefield, the ability goes on the stack and resolves or is countered as normal.

Sometimes a Triggered Ability can be triggered multiple times. In that case, each trigger goes on the stack separately and resolves separately.

If you've got Slogurk out and someone casts Armageddon, first, don't play with that player. Second, Slogurk will count every one of those Lands hitting the 'yard and get a counter for each one. Note sometimes the rules will limit how many times an ability can trigger.

So Lazav can only exile and copy something once per turn, even if you commit multiple crimes, you rapscallion, you.

Sometimes the effect is a "may" ability.

When the events which trigger the ability occur, the effect goes on the stack whether you like it or not; you make the choice upon resolution. So if someone Stifles your Heart-Piercer Manticore's ability, you don't get to make the choice - but they have the option to Stifle it whether you intend to sacrifice something or not.

There are a lot of corner cases and the Comprehensive Rules get into them at 603. The one last thing worth knowing, though, is what to do if multiple abilities trigger at the same time. (This is common on Upkeep and the End Step.) When more than one thing triggers at the same time on your turn, you get to decide the order in which they go on the stack, at which point they resolve in reverse order (because the stack resolves last in/first out).

If you control both of these Creatures and have no Creatures in your Graveyard, you can put Charnel Troll's ability on the stack first, then Lord of the Pit's, then let them resolve. Lord of the Pit will have you sacrifice a Creature, then Charnel Troll will ask you to Exile something - and you've got something there, because you just sacrificed it!

Static Abilities

Comprehensive Rules 113.3d: "Static abilities are written as statements. They're simply true." Most often, this is the form of a permanent on the Battlefield having an effect on the 'field.

As long as Archetype of Courage is a) on the Battlefield and b) still has its abilities, your Creatures have First Strike and your opponent's Creatures don't.

Sometimes Static Abilities are characteristic-defining, which means they determine some part of the permanent in question.

Tarmogoyf's Power and Toughness can go up and down depending on what's in Graveyards, but the ability remains constant.

One final quick thing about Static Abilities - sometimes they affect things on the stack.

As long as you've met the criteria, you can cast Admiral's Order for a reduced cost - that's simply true, even though it's not on a permanent.

I hope this helps you figure out how to resolve some of the complicated game states we can find ourselves in! Please feel free to reach out on social media if you have questions about rules you'd like me to investigate.