In the final rules it's pretty clear: a monster can make multiple attacks only if it has the "Multiattack" action, which always specifies which attacks can be made.
In the case of the Owlbear, it has a Beak and a Claws attack (Claws counts as only one attack). It also has a Multiattack action saying "The owlbear makes two attacks: one with its beak and one with its claws."
If a creature has different attacks but doesn't have Multiattack it has to choose which one to use at each turn.
The base form for the Biped Eidolon is:
Size Medium; Speed 30 ft.; AC +2 natural armor; Saves Fort (good), Ref
(bad), Will (good); Attack 2 claws (1d4); Ability Scores Str 16, Dex
12, Con 13, Int 7, Wis 10, Cha 11
So a base Eidolon will get two claw attacks with a FULL ROUND attack and one claw attack with a normal attack.
The "maximum of three attacks" is for when you start adding things like Bite and Sting. The Eidolon can only have a maximum of 3 attacks (at level 1) no matter how you augment it.
E.g. Tail, Bite and Sting are all 1 point evolutions; so if you took all those as your Eidolons evolutions at level one then it would have 4 attacks (Full round action) Claw, Claw, Bite, Sting. However the maximum number of attacks it is allowed is 3, so one of those attacks would be forfeit.
From your example
Each "attack" counts as an attack; so claw x2 and bite. Each one of those is an "attack", so that's three.
So for a FULL ATTACK the Eidolon would get three attacks (claw, claw, bite)
For a standard action attack the Eidolon would get one attack, this would be either ONE claw or ONE bite as the Bite is listed as a PRIMARY attack - if you only have one attack then you select one of your primary attacks to use.
Best Answer
They get all their attacks every round, and roll to hit separately for them. They can even move and use all their attacks—it's not like D&D 3e, with its "full attack" limitation!
They can split up their multiple attacks against as many engaged opponents as they like. The exception to that is if the description specifically says that they have to attack the same target, but that's pretty rare.
Sometimes you'll see a line like "claw/claw/bite or weapon", in which case they get one set or the other, not all of them: e.g., two claw attacks and one bite, or one weapon attack; not 2 claws, a bite, and a weapon attack.
Yes, creatures with multiple attacks are nasty and dangerous! Make sure you warn players who are B/X novices that such creatures are not to be underestimated.