The FDA defines a vaccine lot number as a code "from which the complete history of the manufacture, processing, packing, holding, and distribution of a batch or lot of drug product or other material can be determined." 
Every vaccine comes with a lot number, like the one in this picture:
TIP: Even though it is required by Federal Law, don't assume that the doctor or nurse is going to keep accurate vaccination records for you. You should always record the lot number in your records and store them in a safe place.
As the FDA points out: "Most drugs consist of pure chemical substances and their structures are known. Most biologics, however, are complex mixtures that are not easily identified or characterized." Vaccines are chemically treated biologics; it is impossible to guarantee that different lots are the same. Lot numbers help manage the safety risks presented by this uncertainty.
Right before you or your child is vaccinated, use Vaccine Lot Search to check US Gov't data for vaccine reaction reports associated with the lot number. You can set VaxCalc™ Alerts on the lot numbers of vaccines you or your child receives. If an adverse event is ever reported to the US Gov't (VAERS) associated with the lot number, you will receive an alert email.
To begin using Vaccine Lot Search, register for a VaxCalc account.
 Code of Federal Regulations, 21CFR210.3
 FDA 101: Regulating Biological Products