Let's suppose you're the US government. You want to encourage certain kinds of behavior in your citizens. You want them to save for retirement, save money for their child's college expenses, buy a house, and maintain an account for medical costs.
Tax deductions are the main incentives that the government uses to encourage these behaviors. A tax deduction is an amount of money that, when contributed toward a certain sanctioned purpose, doesn't count toward your taxable income.
For example, if you earn $60,000 in a given year but contribute $5,000 towards an IRA (a kind of retirement account which is tax-deductible) you'll be taxed as if you had only earned $55,000. This means that you won't pay taxes on that $5,000. In 2017 that money would have been taxed at the 25% tax bracket, so contributing to the IRA and taking advantage of the deduction saved you $1,250 in taxes. Not bad!
The bulk of your taxes will be federal, but most states also collect income taxes.
States have significantly different rates. Some states, like California and New York, have relatively high tax rates—the top marginal rate in California, for example, is 13.3% on income above $1MM. A handful collect no taxes at all; these are Alaska, Florida, Nevada, South Dakota, Texas, Washington, and Wyoming. Ever heard the cultural trope of “New York retiree moves to Florida?” That favorable state tax rate is one of the reasons.
States that have taxes offer their own deductions, just like the federal government does. 529 college savings plans, for example, are state-specific vehicles that are often eligible for state deductions.