The quote is "Render unto Caesar the things which are Ceasar's..." and comes from the Bible, Mark 12:17
Since this is a forum for discussion of language rather than exigesis, I will refrain from discussing the possible deeper contextual implications of the phrase, and instead clarify the semantic meaning.
Please be aware that the quote you have referenced comes from the King James Bible, a translation of biblical scripture that was carried out some centuries ago and therefore using language that can be mystifying to the modern reader. A more up-to-date translation in common use is the New International Version, which renders the quote thus:
"Give to Caesar what is Caesar's and to God what is God's."
This was in response to a question about whether Jews should pay taxes (or instead resist Roman authority.) Please refer to one of the many biblical study guides or websites for a scriptural analysis, or this Wikipedia article for a good primer (thanks J. M. and Scott Mitchell :-)
This particular biblical quote is used in different contexts to mean any one of a number of different things, but in my experience it is most commonly an exhortation to keep the affairs of religion and politics separate.
From the New Oxford American Dictionary:
every other: each second in a series; each alternate: I train with weights every other day
In this case, it is most probably used figuratively, and means “rebooting very often”.
Best Answer
Swept under the rug (or carpet) is an idiom meaning to conceal something that is embarrassing that you don't want other people to know about.
Think about cleaning the floor. You have all the dirt, and are too lazy or out of time to find something to pick it up, so you lift the corner of the rug and sweep it there.