Legislative History generally means two types of research:
1. "Simple" legislative history. What was the federal/state law at particular time? (Ex. What was the NH RSA on LLCs in 2015?) Check the statute's history to find out when it was last changed and then look up the statute from that period. Westlaw and Lexis both have historical state statutes from the late 80s and early 90s (coverage varies by database and state). Historical versions of federal statutes are available on Westlaw and Lexis as well as further back on HeinOnline.
2. Traditional legislative history. This process is finding all the documents associated with a law to determine the intent of the law. These include Hearings, Committee Reports, Previous Bill Versions, Budget Reports, Vote Records, Floor Debates and more. See the resources below for great starting points.
The New Hampshire State Law Library created a useful guide to NH legislative history research here with a companion guide to free online resources available here.
A few points on using these great resources:
Compiled legislative histories are the easiest option - all of the work is done for you!
HeinOnline compiles enormous legislative histories (thousands of pages) for each session of Congress. However, not every bill passed into law has an accompanying legislative history on HeinOnline. The editors at Hein choose the most notable.
Many pieces of legislative history are conveniently linked via Lexis and Westlaw when looking at the USCS or USCA respectively.
This set is GIANT!
It covers material from the early 19th century until 1994. However, such a giant database can sometimes be unwieldy. Simple keyword searches often return thousands of hits. To maximize this resource use advance search to limit by a particular congress, search by citation, or search using specific subjects generated by the database itself.