A seven-year annuity is a form of deferred taxation savings product. In the way that most people use a seven-year annuity, it isn't really an annuity, although it does conform the the legal requirements to be defined as one--that's what provides the tax benefits.
The technical definition of an annuity is a stream of payments that come to you over time. So you might put your retirement savings with an insurance company in return for a fixed yearly or monthly payment until you die.
Since the 1970s a market in deferred annuities has grown. These have the option at the end of converting to a standard annuity but are more often used as a method of delaying or deferring taxes on the money invested.
A seven-year annuity is one of these deferred annuities. You invest with the life insurance company and agree to wait seven years before you can convert to a standard annuity. During those seven years, there is no tax on the growth in your savings.
If you invest directly, you have to pay tax on your returns each year. So you can invest only your after-tax returns for the next year and so on. By using a deferred annuity, you don't have to pay that tax each year, so more is invested each year--the total returns of Year 1 are invested in Year 2 and so on.
What Actually Happens?
At the end of the seven years, most people do not convert a seven-year annuity to a standard annuity. Rather, they take the entire sum and either use it or invest it again. So seven-year annuities and other deferred annuities, while technically annuities, are really more a method of tax-deferred savings.