A bill is a proposed law or a proposed amendment to an existing law, numbered consecutively for each Congress and chamber - H.R. 1; S. 1
Once the bill is passed by Congress and then either 1) signed by the President 2) vetoed by the President, but has the veto overridden by Congress or 3) passed by Congress, but is not acted on by the President for a specified by period of time - the bill becomes a Public Law. Public Laws are numbered by Congress, consecutively in the order they passed - P.L 105-322.
A legislative history of a Public Law can contain multiple bills, all which failed to pass the full Congress. A bill may be revised multiple times before becoming a law.