Valmiki wrote 24000 slokas in Srimad Ramayana.  That is because the 'Gayathri Mantra' comprises of 24 'aksharas', and a thousand verses for each akshara.

 The number '18' is considered very important in Mantra Shastra and a Mantra recited for 18 times or in multiples of 18 denotes something auspicious.  It is for the same reason that Ayyappan Temples have 18 holy steps.
The number 18 signifies fulfilment in life!

There are 18 chapters in Bhagavat Gita, 18 Puranas and 18 'parvas' in the Mahabharata.  Srimad Bhagavatam contains 18000 Slokas as it is the essence of all these.

Sage Vyasa composed Srimad Bhagavatam which is verily the nectar and essence of all the Vedas and Shastras.  It is the fruit borne by the tree of the Vedas.  Sri Suka drunk this nectar from his father and reproduced the same verbatim, without any additions or deletions, just like a parrot repeats verbatim, what it is taught.  And this nectar liberates us from the cycle of births and deaths -- as lucidly put in the sloka -

nigama kalpa tharOr galitham phalam sukamukhAt amrutatravasamyutam |
pibhata bhAgavata rasamAlayam muhur aho rasikAh puvi bhAvukAh  || (1:1:3) 

Secondly, Sri Suka has narrated Srimad Bhagavatam in the most lucid, simple and elegant style - just like a parrot would.  [in reference to the tamil phrase 'kili pillai solvathu pol'  which means, 'as a parrot speaks' ]