This information is based on the Authorised Version (KJV). Other versions differ.
By word count there are two shortest verses, both with 7 words each:
LORD, remember David, and all his afflictions Ps 132:1
His foundation is in the holy mountains. Ps 87 : 1
The longest verse has 45 words:
For in the hand of the LORD there is a cup, and the wine is red; it is full of mixture; and he poureth out of the same: but the dregs thereof, all the wicked of the earth shall wring them out, and drink them. Ps 75 : 8
By character count, three shortest verses share 28 characters:
I said in my haste, All men are liars. Ps 116 : 11
I will love thee, O LORD, my strength. Ps 18 : 1
Unto thee, O LORD, do I lift up my soul. Ps 25 : 1
And the longest has 181 characters:
From men which are thy hand, O LORD, from men of the world, which have their portion in this life, and whose belly thou fillest with thy hid treasure: they are full of children, and leave the rest of their substance to their babes. Ps 17 : 14
Tech note: To generate these statistics, I loaded the text of the KJV into an sql table, one verse per row (from http://www.htmlbible.com/#free). The number of words per verse is LEN(REPLACE(RTrim(TextData), ‘ ‘, ‘–‘)) – LEN(RTrim(TextData)) + 1, and the number of characters is LEN(TextData) – words + 1 (using mssql)