A tax exemption reduces the total tax owed, because a person, or particular source of money is exempt (not counted toward the total). For example, in many countries low-income earners are exempt from paying income tax because paying that tax would be very hard on their finances. Likewise, tax exemptions can be a way of encouraging (or discouraging) particular behaviour, see Pigouvian tax.
A tax deduction reduces the overall amount of income that the government is considers when determining the amount owed in taxes. If a tax payer has a taxable income of $50,000 and they claim a deduction of $5,000 then the amount of taxable income will fall to $45,000.
In the U.S, married couples can claim a "standard" deduction which reduces their tax liability, if the family has dependent children, they can claim additional deductions.
Governments can offer either exemptions or deductions based on what activities they want people to participate in. In Canada, many religious and other charities are exempt from taxation, this helps to encourage people contributing to charitable work.