What do lattice boom and hydraulic boom in cranes mean?

Answer 1

LATTICE BOOM: See first link.
This is where the boom (the portion used to support the load) is made like a lattice ... i.e. interconnected trusses and girders. The cables for lifting the load are run from the base of the boom to the load over pulleys/gears located at the boom's tip. The boom itself is NOT adjustable in length or in angle.

HYDRAULIC BOOM: See second link.
This crane uses hydraulics to adjust either the length of the boom, or the angle at which the boom joints intersect. It looks more like a single beam construction, although its probably closer to a tube type design (hydraulics may be inside or outside the arm). The load is raised/lowered by cables operating at the end of the boom ... not necessarily tied back to the bottom of the boom.