C1V1 = C2V2
When you dilute a solution the amount of solute stays the same, so concentration × volume is the same before and after. C1 and V1 are the stock concentration and the volume of stock you use; C2 and V2 are the concentration and volume you want.
To make 10 mL of 1 M from a 5 M stock: V1 = C2 × V2 ÷ C1 = 1 × 10 ÷ 5 = 2 mL of stock, topped up with 8 mL of water or buffer to 10 mL.
C1 and C2 must be in the same units, and so must V1 and V2.
Dilution factors
A 1:10 dilution is 1 part stock in 10 parts total, such as 100 µL of stock plus 900 µL of diluent. Note that "1:10" sometimes means 1 part plus 10 parts, so check which is meant in a protocol.
Serial dilutions
For very large dilutions, dilute step by step. Six 1:10 steps take a sample from 108 to 102 cells per mL, a dilution of 106, with volumes that are easy to pipette. Mix each tube well before taking the next transfer, and change tips between tubes.
Calculators
The Dilution Calculator solves C1V1 = C2V2 for any of the four values and converts units for you. The Serial Dilution Calculator lists the concentration in each tube and how much to transfer and add.