As long as your IRA dividends stay in your account, they aren't taxed at all. That's the purpose of a traditional IRA: to let your earnings accumulate tax free until you withdraw them in retirement. In most cases, taxes aren't an issue for a traditional IRA until you turn 59 1/2, the minimum withdrawal age. With a Roth IRA, if you wait until the minimum age, there's no tax on earnings at all.
It doesn't matter whether your earnings come from dividends, interest or capital gains -- when you withdraw money from a traditional IRA, it is taxed like regular income, at your regular income tax rate. You can sometimes withdraw contributions tax-free -- if you paid tax on the money when you deposited it, say -- but you always pay income tax on the earnings. The same tax rules apply to beneficiaries who inherit IRA accounts.
If you take money out of your IRA before turning 59 1/2, you pay not only income tax but a 10 percent tax penalty too. Withdraw $2,000 when you're 40, say, and you pay the IRS a $200 penalty. The law allows you to make early withdrawals without a penalty in certain cases. These include buying your first home, paying for tuition or covering major medical expenses. If you deposit too much in your IRA and have to take money back out, there's no penalty on those withdrawals either.
If you have the IRA administrator transfer your money to another traditional account, there's no tax on the rollover. You can roll the money over yourself, but if you don't complete the transfer in 60 days, the IRS taxes it like an early withdrawal. Another special case is when you withdraw money from your IRA to give to charity. Provided you're over 70 1/2, there's no tax on withdrawals that become donations.
Your IRA account administrator will send you a 1099-R form stating how much you took out of the account each year. Report the money on your 1040, adding it in with your other income. If some of the money was withdrawn tax-free, the 1099 will separate that out. You don't have the option to choose whether withdrawals are tax-free or not. If, say, 10 percent of your IRA can be withdrawn without tax, the IRS simply counts 10 percent of every withdrawal as tax-free.