Good, so,
are you can understand first several strings from this error?
Seems error already in english? :)

so, read:
Database query error
Query:
SELECT COUNT(`ID`) AS `Count` FROM `RayLivecamChannels` WHERE `Public`='true'
Mysql error:
Unknown column 'Public' in 'where clause'

last string what mean?
That you haven`t column 'Public' in your `RayLivecamChannels` table, not more :)
And seems that it anybody mod, So you must create this field for this table, understand? :) all simple