So, this is how poisons (and most other afflictions) work:

• You suffer the effects every time you fail a save
• The frequency tells you how often you make a save, once the onset time has passed
• If there is no onset, you make a save immediately
• The poison is finished when the conditions for a cure are met, or the duration has elapsed

For black adder venom, there is no onset, and the frequency is “1/round for 6 rounds”. The cure is “1 save”. That means that:

• Since there is no onset, when you are hit by the poison, you make a save immediately.
• If you fail, you take 1d2 Con damage. If you succeed, you've met the conditions for a cure, and so the poison is now out of your system.
• Because the frequency is “1/round for 6 round” you repeat this once per round until you either make a save (and thus are cured) or 6 rounds have elapsed. Every time you fail a save, you take the damage. This damage can be healed by any of the normal means of recovering strength (restoration spells, resting, etc.)

The second example of a sting is almost identical: the frequency and cure are the same, but the effect is being sickened. The effect is that you're sickened until the poison is up, which is either the first time you make a save or after 6 rounds. (Note that if someone e.g. used magic to heal you of the sickened effect, it would return the next round. It's 1 round of sickened applied 6 times in a row, not 6 rounds of sickened applied once.)

This all follows from the general rules for afflictions -- a category which include poisons and diseases. Diseases tend to have a longer onset time and lower frequency than poisons, but the basic mechanics are the same. If you still find yourself confused, take the time to read through the afflictions page I linked to.

As BESW said, don't expect the game system to model reality.

That said, if you need to figure out stats for potato tea...

• It's an ingested poison. As the SRD says,

  Ingested poisons are virtually impossible to utilize in a combat situation. A poisoner could administer a potion to an unconscious creature or attempt to dupe someone into drinking or eating something poisoned. Assassins and other characters tend to use ingested poisons outside of combat.

  So, this really isn't likely to be a major combat advantage. In my view, a cheap and accessible plot-device poison isn't really a big deal. Also, how much exactly do you have to trick the target into drinking? From the sound of it, potato tea isn't nearly as potently concentrated as the expensive fantasy poisons in the game book.

• What's the save DC? Something that calls for a DC 12 Fortitude save, for example, is still going to mess up a lot of low-level characters, but most powerful people will be pretty much immune to it.

• What's the actual effect in game-mechanical terms? In the real world, it's pretty lethal, but lethal in the real world doesn't necessarily translate to "save or die." A poison that does 2d12 damage could kill a low-level commoner. Whereas a powerful character is just going to vomit it up but survive pretty much unscathed.

I'm not saying you should block the players by going out of your way to make it weak and useless. But don't make a game action super-powerful just because it could kill someone in reality. People were killed with regular ol' run-of-the-mill arming swords all the time. Yet they still do only 1d8 damage in D&D. Players characters and antagonists in D&D very quickly become more like action-movie heroes (or wuxia/super heroes, even) than real-world people as they attain levels or Hit Dice.