Any other effects of that damage, such as life gain from lifelink, still apply.

Multiple instances of toxic are cumulative. For example, if a creature has toxic 2 and gains toxic 1 due to another effect, combat damage that creature deals to a player will cause that player to get 3 poison counters.

If a creature with toxic deals combat damage to a creature or planeswalker, or if it deals noncombat damage, toxic has no effect and no player gets poison counters.

Damage dealt by a creature with toxic grants the same number of counters regardless of how much damage is dealt. Notably, if a replacement effect modifies the damage in some way (such as that of Gratuitous Violence), the number of counters given remains unchanged.

A player with ten or more poison counters loses the game. This is a state-based action and doesn't use the stack. In other words, it happens immediately and players can't respond to it, just like a player losing the game due to having 0 or less life.

Toxic doesn't change the amount of combat damage a creature deals. For example, if a 2/2 creature with toxic 1 deals combat damage to a player, that creature will deal 2 damage. The results of that damage are the player loses 2 life and gets a poison counter.

Conversely, replacement effects that apply to the number of counters put on a player can modify the counters placed this way. For example, Vorinclex, Monstrous Raider's last two abilities can apply to counters placed this way.