Free attacks or extra attacks are the attacks that can be made in addition to those granted by a character's base attack bonus and dual-wielding. (So a case such as whirlwind attack, where attacks are made instead of regular attacks does not fall into this classification.) They are usually made with the main hand weapon, and are not reported on the character sheet. In most cases, sources of free attacks describe them as "at full base attack", but this is only accurate when there is a single free attack in a given round.
For the most part, free attacks form their own progression, starting at full base attack and proceeding at -5 (even for those qualifying for UBAB), and they occur between the regular main-hand attacks and the off-hand attacks. However, there are times when a free attack is inserted in the middle of the main- or off-hand attack sequences. If this happens, the attack progressions can appear rather strange, sometimes to the benefit of the attacker, sometimes to the detriment thereof (compared to what would take place if the free attack was inserted between main- and off-hand attacks). However, the end result is always better than the case without the free attack.
When a free attack is inserted in the main-hand sequence, a main-hand attack gets "promoted" to either full base attack (if this is the first such promotion) or -5 off the previous promotion's base attack, and a "true" free attack is added to the free attack progression to make up for the lost regular attack. When a free attack is inserted into the off-hand sequence, it is essentially converted to an additional off-hand attack, following the usual progression. (So with improved two-weapon fighting and a free attack inserted into the off-hand sequence, a character could end up with three off-hand attacks, with the second at -5 and the third at -10.) These insertions only occur when the free attack is generated by some combat action, such as an attack of opportunity.
Free attacks in the usual sequence (between main- and off- hand attacks) ignore dual-wielding penalties.
Free attacks applied to users of creature weapons are implemented as additional attacks applied at end of the attack progression and continue the same BAB progression rate as the class dictates (-3 for monk, -5 for all other classes), ignoring the normal "full base attack" effect.