My students know this means "holding all other factors constant." This is part of what makes economics so powerful. For example, we know many factors influence how much of a product you will buy (like income, tastes, ect.) But if we assume that all of those are held constant, we can say that when price falls, ceteris paribus, that the quantity consumers wish to purchase rises. This is known as the Law of Demand. It allows us to describe how markets behave (along with the Law of Supply).
We also know how a market will be affected by a tax or a regulation since we will hold all other factors constant. We can make a reasonable prediction using supply and demand. For example, if sellers have to collect a dollar every time they sell one unit and give it to the government (an excise tax), we can easily show that price will rise and the quantity bought and sold in the market will fall. We can also determine how much of the tax the buyer will pay and how much the seller will pay (and it is not always an even split). Now that's magic.
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