Things you need:
Tanqueray No. 10 Gin
Noilly Prat Vermouth
A sharp knife
A cocktail shaker
Fresh ice, preferably made with filtered, distilled, or spring water
A cocktail glass, about 12cm (4.75") diameter at the top
Chill the cocktail glass in the freezer long enough to bring it down to well below freezing.
Keep the vermouth in the fridge, and the gin in the cupboard (not the fridge or freezer). Noilly Prat comes in 1L bottles; you will almost certainly end up dumping some of it, as it only keeps for so long (some say a month, I figure that in a fridge, you can probably stretch that to 3 months).
(Olive lovers will probably find that Tanqueray Gin isn't ideal for a martini with olives; Hendrick's is supposed to be better for that.)
Before pouring anything, prepare the twist: Cut a 6mm (1/4") slice of lemon from close to the middle, then cut out the flesh from inside, leaving a ring of lemon peel; cut it in one place to make a strip and set it aside.
Measure out the gin; I use one shaker capful, which is probably a couple of ounces. Pour the vermouth into the cocktail shaker, measuring with a suitable implement. I use a small soup spoon, and pour in about 2.5 spoonsful. My Gin:Vermouth ratio is about 7:1. Opinions vary on what the optimum ratio is; experiment!
Half fill the shaker with ice, then pour the gin over the ice. Cover the shaker, and shake with moderate force 30 times. Take the glass out of the freezer, and strain the martini into the glass, shaking out the last drops. There should be about 2cm (most of an inch) between the top of the liquid and the top of the glass.
Squeeze the lemon peel into a U shape in its narrow direction, flesh side in, over the glass, sliding your fingers down the peel (this takes practice!). Then, twist the peel lengthwise, keeping a bit of tension on it, until it is fully twisted, let it spring back, and put it into the glass so that one end is in the bottom of the glass and the other end hangs over the edge.
Hold the glass by the stem so as not to warm it up prematurely, and enjoy with care. This packs a fair bit of punch, so don't drink it too fast, or if you need to drive anywhere soon!