An annuity is where you hand over all or part of a pension pot to an insurance company in exchange for a regular monthly income. The amount they give you depends on how much money you have in your pension pot, or choose to invest, along with various calculations about your state of health and life expectancy, as well as what level general interest rates are at, at the time you start the annuity.
The older you are and the poorer health you are in, the more they will give you per month in exchange for your investment (as they reckon there is a fair chance of you dying before you have cost them money!). If you beat the average and live to 120 then you are quids in!
Annuities also tend to pay more when interest rates generally are high, and vice versa. In recent years they have been reckoned to be poor value as interest rates were very low in the UK, but have come back into favour in the last couple of years.
An annuity can either be a flat rate per month (in which case its effective value to you decreases as time goes on due to inflation), or increase by a percentage each year to try and keep pace with inflation. The starting amount you get per month with an increasing annuity is less than you would get with a fixed rate one.
The downsides to an annuity are that once you have set one up you can’t normally change it or withdraw your money, and usually it expires with your death i.e. there will be nothing left for the beneficiaries of your will (though you can get an annuity that is transferable to a spouse on your death). There are other types but those are the basics as I understand them!
“Draw-down” on the other hand means you leave your money invested with the pension provider, and just take money out either on a regular basis or as and when you need it - the frequency and amount withdrawn is up to you.
The advantage of this is the money stays invested and should continue to increase as time goes on, but you are also exposed to the vagaries of the stock market so if the fund does less well for a time you may have to reduce your withdrawals for a while, and essentially pace yourself so that the pension pot doesn’t run out before you die.
There is a rule of thumb (worked out by a US analyst called Bill Bengen) that said you should be able to “draw down” 4% of your pot each year and it would not run out - this was based on analysis of investment performance from 1926 to 1976.
Obviously you could take out less (say 3%) if you wanted to be cautious, or a higher percentage if you have other sources of income or felt adventurous (or are not planning in leaving any money to the offspring etc.!)
With drawdown if there is money left in your pot when you die it goes to your heirs, as per your will (and French law!)
That’s my understanding of it anyway - can you tell I have been researching this recently?
@Dave_Lawson do I get my Blue Peter financial adviser badge? 
that I could only keep my pot in drawdown until I’m 70, when buying an annuity would be obligatory. My IFA is doing a good job (I think) as I’m taking about 8% keeping the capital stable. If you’re interested I can give you his details- he’s in Banstead so reasonably local for you.