Merriam Webster defines denomination as “a value or size in a series of values or sizes.”
The most common usage outside of religion is to describe money. Our bills come in the one dollar denomination, the five dollar denomination, the ten dollar denomination, and so on. A five dollar bill is different than a ten dollar bill but they both are in the series of dollar bills.
Denomination comes from the Latin word “denominare”. In Latin, the prefix “de-” was used in various ways but one of those ways was to say off, away, or from among. And “nominare” in Latin was to name. Put them together and “denominare” is to name something from among other things.
Suppose you are hiring for a position and you need to separate out the various applicants. You could denominate one group as college educated and another group as non college educated. Or you could denominate one group as ten plus years of experience, another group as five to ten years of experience, and a third group as less than five years of experience. By naming these groups, you are separating them out from among the overall pool of applicants.
A church is a denomination when it follows a defined template of rules or practices that makes it unique from other churches. Usually this means there is some kind of centralized organization for all the groups that follow this template.
We consider ourselves non-denominational because we do not agree with the idea of different sets of rules and practices for Christians. We believe the only acceptable set of rules and practices for Christians are the ones laid out in the bible and that a Christian should just be a Christian, a follower of Christ and a member of His Church.
The apostle Paul dealt with denominationalism, or at least something very close to it, in his first letter to the Corinthians.
The Corinthians Christians were apparently starting to divide themselves by choosing a leader from among at least three men who taught the gospel -- Paul, Apollos, and Cephas (Peter). But Paul condemned this idea. Neither Paul, Apollos, nor Cephas died for their sins. Jesus Christ did that and Jesus Christ is not divided.
Jesus Christ is the Head and Savior of the Church, which is His body (Ephesians 5:23; 1 Corinthians 12:27). And the bible says that there is only one body.
See also: Romans 12:4-5; 1 Corinthians 12:12-13