First off, it's good to know you don't attack a creature like in Yu-gi-oh, you attack a player or planeswalker, and a creature may be declared as a blocker.
When a creature with first strike is attacking, an extra combat damage step is created to accommodate said creature, called the First Combat Damage Step, followed by the turn's regular Combat Damage Step.
Creatures with first/double strike will deal their combat damage in the first combat damage step, while creatures without either or with double strike will deal their combat damage in the regular combat damage step.
Ancestor's Chosen will deal it's damage first in the First Combat Damage Step, before Baleful Eidolon ever gets a chance to deal combat damage in the regular combat damage step. In order for deathtouch to apply, the creature with it must deal damage, but in this case, that said creature will be dealt lethal damage and die before it gets the chance to. Your Ancestor's Chosen will survive to fight again.
January 20, 2016 3:34 p.m.
Something worth noting:
Outside of the First and Regular Combat Damage Steps, first strike is not applied whatsoever. If Ancestor's Chosen and Baleful Eidolon were made to fight each other, (perhaps from Pit Fight), they would both deal damage to each other equal to their power. First strike does not apply here, and the damage will be dealt simultaneously. Baleful Eidolon will die due to lethal damage, and Ancestor's Chosen will die due to lethal damage as well. (1 damage of deathtouch is considered lethal damage).
January 20, 2016 3:38 p.m.
I realize that creatures attack the player, I just figured it was easier to say the creatures were attacking each other...
But thank you for answering my question and clarifying the other instance!
January 20, 2016 3:44 p.m.
No worries. Better to clarify any possible misunderstandings for other people looking at the question. :)
Also something to know:
If a creature with first/double strike has dealt its damage in the first combat damage step, but loses the ability before damage is dealt in the regular combat damage step, that creature will not deal any damage in the regular combat damage step.