How do poison counters work and how would a player get them?
All you really need to know is that you lose the game if you get 10, and that they're nearly impossible to remove. Generally if you see a card that talks about poison counters, it'll be a card that gives them, so this should never really be confusing - if the cards don't mention them, you don't have to worry about them.



By far the main way to get poison counters is from creatures with Infect, like Blighted Agent:


  Infect (This creature deals damage to creatures in the form of -1/-1 counters and to players in the form of poison counters.)


There's was also an ability called Poisonous, but it's only on a couple cards.  Virulent Sliver gives all Slivers Poisonous 1:


  All Sliver creatures have poisonous 1. (Whenever a Sliver deals combat damage to a player, that player gets a poison counter. A player with ten or more poison counters loses the game.)


And Snake Cult Initiation is an aura that gives the enchanted creature Poisonous 3. Marsh Viper essentially has Poisonous 2, but spells it out.

There are several more cards that just explicitly give a player poison counters.