Needle conversion chart
Yarn standards chart
These charts will tell you how to convert UK needles to US sizes.
Worsted weight yarn is the standard, most common weight yarn, whose gauge is about 20 st per 10 cm.
In order to get your squares to be the same size as the pattern says, you need to be knitting in the same gauge. Even if you use the correct size needle, that doesn't always happen. People knit at different tensions. If your square is turning out too big, you can try with smaller needle; too small, try a larger needle. The gauge for these squares is 5 st per inch.
Very often when you cast on stitches, they don't measure the same as what the knitted piece will be, so you need to knit a few rows before you get a better idea of what it's true size will be.
I hope this gets you started, anyway.