You almost always need a Full-round Action to take multiple attacks
Natural Weapons are not an exception
As a standard action, you can use any of your natural weapons to make a single attack. You cannot make attacks with multiple natural weapons, not even paired weapons like claws. If a weapon is “secondary” (it will say so in its description), it makes the attack at −5 even if it’s your only attack.
As a full-round action, you may attack once with each weapon. Only one weapon or pair of weapons can be “primary;” even if you have more than one weapon or pair of weapons that claim to be “primary” in their description, you must pick one or one pair to be primary, and all others are secondary. Secondary natural weapon attacks are taken at a −5 penalty.
This means your attack routine with two claws (primary pair) and a tail attack (secondary) is +0/+0/−5 at BAB +0.
There are some (rare) ways to attack more than once as a standard action, and some of them are compatible with natural weapons. These will explicitly state this exception to the usual, however.
Natural Weapons and Manufactured Weapons use totally separate rules
Natural weapons never receive iterative attacks, and there are no cumulative −5 penalties, only a single −5 penalty applied to all secondary natural attacks. If you have the Multiattack feat, all secondary natural weapons attack at a −2 penalty instead of a −5 penalty.
None of the rules about manufactured weapon attacks (iteratives, two-weapon fighting, Flurry of Blows, etc.) apply to natural weapons.
Thus at BAB +20, that same pair of primary claws and secondary tail slap attacks at +20/+20/+15, or +20/+20/+18 with Multiattack. If you added a pair of secondary Tentacle attacks, that would be +20/+20/+15/+15/+15 without Multiattack, or +20/+20/+18/+18/+18 with.
Unless otherwise specified by the natural weapon, primary natural weapons add your Strength modifier to damage, while secondary natural weapons add half your Strength modifier to damage.
You can mix manufactured weapons and natural weapons.
When you do so, you gain all your normal manufactured-weapon attacks (including iteratives), plus any natural weapons you have, each as a secondary weapon, with the caveat that you cannot use any natural weapons that are being use to hold a manufactured weapon. I.e. if you hold your sword in one hand and attack with it, you cannot use the Claw attack from that hand. If you instead two-hand a Great Axe, then you cannot use either Claw attack.
When you do this, you can mix natural weapons with full-attacks that use the two-weapon fighting combat option or Flurry of Blows class feature.
Example: Longsword, pair of claws, and tail slap
Thus if you had a Longsword, a pair of Claws, and a Tail Slap, your attack routine would be +0/−5/−5 (sword, claw, tail) for BAB +0. Multiattack improves this to +0/−2/−2. At BAB +20, it’s instead +20/+15/+10/+5/+15/+15 (sword, sword, sword, sword, claw, tail) or +20/+15/+10/+5/+18/+18 if you have Multiattack.
Same character with dual kukris and the Two-Weapon Fighting feat
If you instead have the Two-Weapon Fighting feat, and use that option with a pair of light weapons (call it a pair of Kukris), you attack at −2/−2/−7 (kukri, kukri, tail), since both hands with Claws are being used to attack with Kukris, and the Tail is always secondary when used with manufactured weapons (hence −5; note that Two-Weapon Fighting does not penalize it, however). The Multiattack feat would reduce the penalty, so the tail would be at −2 (+3 attack for one out of three attacks is not a great use of a feat, but just for completion’s sake).
At BAB +15 and with the Improved and Greater Two-Weapon Fighting feats, you get +13/+13/+8/+8/+3/+3/+10 (left kukri, right kukri, left kukri (2nd), right kukri (2nd), left kukri (3rd), right kukri (3rd), tail).
It's based on a toy.
And that's all most sources will give you, because they're drawing from an article (Ed Greenwood's "Ecology of the Rust Monster" in Dragon #88, later quoted by another article of the same name in issue #346) which is actually about the rust monster, and only passingly mentions the owlbear. In fact, the original quote is so vague it's hard to tell if they actually mean to say the owlbear shares the same origin as the rust monster (a plastic toy).
But Tony DiTerlizzi gives us photos of the actual toy which inspired this freaky creature, and so here it is, the original owlbear:
It comes from a pack of early 1970s cheap plastic toys depicting "prehistoric animals," most of them entirely made up. Look, you can see the bulette and the rust monster, too:
The origin of these toys is from Hong Kong, where they made these toys to resemble monsters from the 1960's famous UltraMan TV show. source
Best Answer
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.