In Article I, section 9, the Constitution says:
No Capitation, or other direct, Tax shall be laid, unless in Proportion to the Census or enumeration herein before directed to be taken.
A "capitation" is also called a head tax; it means you tax $X for every person. The meaning of "other direct Tax" is less clear; the historical understanding in the late 19th century was that it didn't apply to income. However, one thing that is clear is that it does apply to a property tax: if the federal government seeks to tax you on the value of your real or personal property, it must apportion the tax among the states on the basis of population. If Virginia has 2.5% of the US population and New York has 6.25%, then the total tax money collected from Virginia must be 2.5% of the total tax money collected nationwide, and likewise for New York; the total money collected from Virginia must be 40% the amount collected from New York.
This doesn't mean Virginia's rate is 40% of New York's. If Virginia's land had a total value of 40% what New York's did, then the rates would actually be the same. But the ratio between tax collected from the two states would have nothing to do with property value; it would be just based on population. This is a rather convoluted and silly system; one theory I've read suggests that part of the point was to discourage these sorts of taxes unless they're truly necessary (and if they do happen, to collect the most money from the states with the most congressmen that could vote on the bill). But it's unworkable for almost any actual tax that's not a head tax, and head taxes are very much out of fashion.
Most taxes have been considered to not be direct taxes. Income tax was generally not considered direct either. However, an 1895 Supreme Court case (Pollock v. Farmers' Loan & Trust Co.) found that a tax on income which was derived from property (rent, dividends, interest, etc.) was effectively a tax on the property, and so was a direct tax. This would apply to only a subset of income (wage income wasn't included), but it's a reasonably significant subset.
Many people didn't particularly like the decision; it pretty much said "the kind of income rich people make can't be taxed by a practical method, only this impractical apportionment system." The Supreme Court can be overruled via constitutional amendment; that's what the 16th Amendment did. It clarified that Pollock was no more, and that income tax didn't need to be apportioned no matter where the income came from. It was enacted precisely because apportionment is utterly impractical.