This is a tricky one, but to sum it up, nucleophiles tend to be less basic.
Nucleophilicity refers to "how much" a reactant wants to "find" a positive charge. In other words, it measures the instability of the reactant's negative charge.
Basicity refers to "how much" a reactant wants to "find" a Hydrogen ion.
In both of these, the underlying drive is the same--both the base and the nucleophile want to stabilize their negative charge with a positive one.
So, the more unstable the negative charge of a reactant is, the higher its nucleophilicity AND its basicity.
The defining difference, then, is how a reactant behaves in the presence of a substrate that contains an electrophile or an abstractable hydrogen
A strong base will have such a great thermodynamic instability (great energy--such as #H^-# or hydride) that it will attack a protic hydrogen to form #H_2#
A good nucleophile, then, is not as basic and is more likely to be sterically unhindered. Consider #CN#. It will tend to act as a nucleophile and attack an electrophile
A reactant can be a good nucleophile and a good base and act as either.
Consider # HO^- # or hydroxide. Depending on the conditions, it can act as a base and turn into water, or it can attack an electrophile in an #SN_2# fashion
Consider #Br^- # which is a base and a good nucleophile, It will tend to act in an SN2 fashion
As you can see it is very nuanced, and you have to consider the whole reaction when determining nucleophilicity and basicity.
I have to sign off now, but feel free to ask more questions or add more detail.
Hope it helps!