Counters are a specific object in the game, usually represented by a physical token (glass beads, tiny dice, pennies if you're desperate, &c.) so you can keep track of them.
+1/+1 counters are a specific kind of counter with a baked-in ability: modifying the power and toughness of a creature they are attached to.
Only things specified as "counters" in the game rules or card text are counters.
All of these things are not counters and do not involve counters:
- Your creature's base power and toughness, and effects that set them to specific values.
- A creature's power and toughness that gets set by the card text while the value in the corner just says "*/*."
- A card that modifies a creature's power and toughness, such as "target creature gets +3/+3 until end of turn" or "all white creatures get +1/+1."
The only times you're likely to be interacting with counters in Magic without a card being super explicit are planeswalker loyalty and a handful of specialized set mechanics:
- Infect and Wither turn damage into -1/-1 counters.
- Suspend and cumulative upkeep both use counters to keep track of their "ticks."