OM is “Operations Manual”, the printed rules that come with the Avalon Hill version of Axis and Alllies
LHRT is Larry Harris (the developer of the Revised game) Tournament Rules. Version 1.3 is in the works (clarifying multi-national naval forces rules) and you can find links to LHRT (the most common verson of the rules for use in online play) by searching these forums for either LHRT or Larry Harris
Avalon Hill’s official rules are the OM as modfied by their online postings of manual erata (corrections of things like the OM saying that SBR damage is a roll of 1d6, or the PC value of the territory bombed, whichever is GREATER)
Most folks use LHRT for online play, since those are the msot clear, have the fewest amiguities, and are widely accepted since they come from the game designer.