Why is NaCN a stong nucleophile but a weak base? Why is NaOCH3 a strong nuc and strong base?
Or how are you suppsed to know that NaSH is a stong nucleophile but a weak base?
Those are good questions and I should point out that the answer is not obvious. Secondly organic chemistry is fundamentally an experimental science, and some things are simply so multivariable that there is no way you could figure them out from first principles without actually doing experiments and measuring things. Nucleophilicity is one of those things. There are multiple variables and they can work in opposing directions. So if it seems contradictory in places, you're right, but it's ultimately based on empirical measurements of reaction rates.
Pibond is absolutely correct in saying that basicity is a thermodynamic phenomenon and nucleophilicity is kinetic.
In basicity we're talking about affinity of the lone pair for a proton. The base strength is related to the thermodynamic stability of the conjugate base.
Nucleophilicity is a lot more of a multivariable issue, and much tougher to untangle. Essentially what you're looking at is the interaction between a lone pair and an orbital (essentially an antibonding orbital) - for example the sigma* in the SN2 and the pi* in 1,2-addition to carbonyls. We're ultimately talking about orbital overlap, and measuring the "strength" based on reaction rate. Orbital overlap meaning the HOMO of the nucleophile and the LUMO of the electrophile.
There are a few factors in play.
1) base strength: strong bases tend to be strong nucleophiles. The key word is tend to be - if there's a competing acid-base reaction that can happen, it almost always will.
2) size. CN- for instance is a very small nucleophile and all else being equal will have smaller steric interactions (key phrase - all else being equal - it never is).
3) "polarizability" - as one goes up the periodic table the valence electrons are less tightly held. The HOMO of the nucleophile is increased, which leads to a stronger interaction with the LUMO of the electrophile. For example this is why HS- is a good nucleophile despite being a poor base.
4) presence of adjacent donating groups - hydrazine and hydrogen peroxide are much stronger nucleophiles than ammonia and water, for instance. The reason is again a higher HOMO due to the pi-donating atoms next door.
Again if these seem like they conflict sometimes, they do. There are multiple factors in play. This is why it doesn't get discussed in detail until late undergrad in a chem major program.
Pibond's gave an excellent nuts and bolts test-oriented summary. Bottom line for the MCAT is learn that, and remember the 3 or 4 prominent examples of "strong nucleophile weak base - like thiols/deprotonated thiols, hydrazine/hydrogen peroxide, cyanide, etc. Should be fine.
hope this helps - James