You may owe gift tax when you give away an item of value that isn't a payment for a purchase. As the donor, you're responsible for paying the gift tax --- the recipient doesn't owe tax on a gift. The recipient only owes income tax if the stock is payment for services rather than a gift.
You can give a gift that's free of any gift tax up to a threshold amount established by the Internal Revenue Service (IRS). For 2010, the threshold is $13,000 to any person during a calendar year. This annual exclusion from gift tax is adjusted every year for the cost of living. You don't owe gift tax when you give less than the threshold maximum to any individual during a year. Exclusion from gift tax also applies if the recipient is your spouse, a political organization, a charity or someone who uses the gift to pay tuition or medical expenses. The amount given for a gift of stock is the value on the date it's transferred.
Giving stock to a charitable organization doesn't result in a gift tax regardless of the stock value. In fact, you're entitled to a tax deduction for the fair market value of stock given to a charity. You must itemize deductions on your tax return to deduct the charitable donation.
The fair market value for each share of stock traded in an active market is the average of the highest- and lowest-quoted selling prices for the day. If the gift is made on a date when the stock market is closed, value is determined by averaging the valuations on the nearest trading days before and after the gift date.
There's no tax consequence on the gift date for an increase in value from your cost. Tax is paid on the capital gain only when the recipient sells the stock.
The recipient of your transferred stock usually obtains your cost basis. However, the basis used to calculate the future capital gain of the recipient also depends upon the stock's fair market value on the gift date. When the fair market value is equal to or greater than the donor's cost basis, the recipient's basis is simply the donor's basis. The recipient's basis is increased by any gift tax paid.
The recipient may have a different basis if the fair market value at the time of the gift is less than the donor's basis and the recipient's sale results in a loss. In that case, the recipient uses as basis the fair market value at the time of the gift --- not the donor's basis.