Short answer: 1 extra spell for specialization per spell level. So, one extra first level spell, one extra second level spell.
At third level, a universalist gets 4 cantrips, 3 first and 2 second level spells per day. A specialist gets 4 cantrips, 4 first and 3 second level spells per day. Cantrips are unique in that they do not get adjusted by the specialist rule.
Long winded explanation:
How do schools work? Well say you were going to focus your studies in the Transmutation school. As a specialist, you are known as a Transmuter. You can go general or focused for the following three choices:
A) general ~ Physical Enhancement at 1st, Telekinetic Fist at 1st, Shape Change at 8th for powers (Core p.82)
B) Enhancement Focus ~ Physical Enhancement at 1st, Enhancement at 1st, Perfection of Self at 8th (Core p82, APG 146)
C) Shapechange Focus ~ Physical Enhancement at 1st, Battleshaper at 1st, Shape Change at 8th for powers (Core p.82, APG 147)
Since you are a specialist, you choose two schools that you will for the most part ignore. These are your opposition schools. Many Transmuters choose Necromancy and Enchantment for their opposition schools. Divination is not allowed as an opposition school.
With an 18 INT, at third level as an example, you will have:
A) four cantrips which can be from you specialty school or your normal schools. If they are from your opposition school, you need to spend two cantrip slots to memorize the one opposition school spell.
b) First level spells: Three first level spells that can be from any school, (again, opposition spells cost 2 slots) and one spell that must be from your specialization school.
2 from the wizard's table, 1 from high Int, 1 for the specialization slot
C) Second level spells: Two first level spells that can be from any school, (again, opposition spells cost 2 slots) and one spell that must be from your specialization school.
1 from the wizard's table, 1 from high INT, and 1 for the specialization slot
D) And then you have a bonded object. Once per day, you can cast a spell you haven't memorized from any spell level that you are able to cast as long as it is written in your book. It is you wildcard spell.
So, a typical transmuter with 18 INT would have
Cantrips: Detect Magic (div), Acid Splash (conj), Prestidigitation (universal), Dancing Lights (evocation)
First lvl: Mage Armor (conjuration), Magic Missile (evocation), Vanish (illusion), and Enlarge Person (transmutation) in the Specialization slot
Second: Glitterdust (conjuration), Scorching Ray (evocation) and Spider Climb (transmutation) in the specialization slot
And when an orc surprises him and chases our wizard down the hallway, he can cast a Flaming Sphere as his bonded object spell to block the hallway as he runs away. (wildcard spell for the day)
Spells cast from your specialization school also get a +1 to their DC check to resist the spell. That raises the reflex or will saving throw they need to make by one.
Transmutation has a lot of useful utility spells. So, it is easy to fill the specialization slot
Conjuration has a lot of summoning and teleport spells.
Divination has weak spells, but very strong extra powers for specializing.
Evocation is you energy direct damage spells.
Enchantment is you charm and control school.
Illusion, self explanatory
Abjuration is you protection spells
Necromancy has few spells to choose from
The best way to figure out what kind of wizard you want to be is to look at the spell lists and see what you would likely memorize and use every day. Then look at what schools those spells fall into the most. From there, look at the school general powers granted or the focused powers from the APG. Choose what enhances your spells the best.
So, if you want to toss fireballs a lot, you go become an Evoker specializing in the Evocation school with the Admixture focus. That way, you can throw fireballs that do lightning damage for the "cool factor" when you are fighting a fire resistant creature.
If you want to charm people and creatures, Enchantment is the path. etc.