A triggered ability happens on its own, whenever a specific event occurs — no player chooses to trigger it. An activated ability is the opposite: a player chooses to pay a cost to get an effect, any time they have priority.
An "ETB" (enters-the-battlefield) ability is just a specific kind of triggered ability — one that triggers when a permanent enters the battlefield. It's not a third category mechanically, just the most common trigger condition players talk about by name.
If you're choosing to do it and paying for it, it's activated. If the game just does it because something happened, it's triggered — ETB is simply the most common trigger of all.
— the fastest way to tell them apart
Both ability types are defined in rule 112, with full detail in rules 602 and 603.
Comprehensive Rules · Rule 112.3b
"Activated abilities have a cost and an effect. They are written as '[Cost]: [Effect.]' A player may activate such an ability whenever they have priority."
Comprehensive Rules · Rule 112.3c
"Triggered abilities have a trigger condition and an effect. They are written as '[Trigger condition], [effect],' and include (and usually begin with) the word 'when,' 'whenever,' or 'at.'"
Because they aren't cast or activated, triggered abilities can trigger even at times when it wouldn't be legal to cast a spell or activate an ability — the game doesn't ask permission first.
How to take advantage of it
ETB effects reward "blinking." Since an ETB trigger fires every time a permanent enters, exiling and returning it (flickering it) retriggers the ability — a whole archetype built entirely around this fact.
Activated abilities are a resource, not a guarantee. Unlike triggers, you have to remember and choose to use an activated ability — it never happens for you automatically.
Countering abilities is different from countering spells. Cards that counter a spell generally can't stop a triggered or activated ability — you need an effect that specifically says it counters abilities.