Heavy and light, as you might expect, deal in terms of weight. Those terms have been applied to many different kinds of weapons. But as it pertains to land forces, and specifically infantry, heavy/light generally refers to the weight and type of equipment a soldier carries.

Heavy Infantry, generally, are the infantry which make up the main battle line of an army. During ancient times, they would be the Legionaries, Hoplites, and other similar troops. They would have the heaviest armor, potent weapons, and sometimes large shields. Their job, which was and is very dependent on what kind of weapon a soldier has, was to fight the enemy army, stand in the line of battle, and do the bulk of the fighting.

Light Infantry, on the other hand, generally carry less equipment, wear lighter armor, and have lighter weapons. Light infantry units generally fought in front of the many body, or along the flanks. Their job was to scout enemy positions, drive off the enemy light infantry, and harass the enemy main body in preparation for an attack. Their "classic" form would be the javelineers, peltasts, and archers.

However, as infantry technology and the tactics of infantry warfare changed, so to did the role of both heavy and light infantry. At some points, the difference between heavy and light infantry was almost non-existant. During the Napoleonic Wars, for example, light infantry companies and battalions could be found in almost every army. Some were equipped with unique weapons like rifles, while other simply fought with the same weapons as the heavier line infantry, but did so in an irregular style. Light infantry units were also regarded more highly than that of a regular unit, and as such they were often treated as a more elite unit within the army.

Today, the differences between light infantry and heavy infantry is not that great. They use the same weapons, carry the same packs, and fight in much the same fashion as "heavy" infantry (motorized, mechanized, and armored). Rather, the difference is, mainly, in transportation and in status. Light infantry of the 20th century can be airborne (dropped from a plane) or airmobile (carried in a helicopter). They are also some of the elite operators within an army, eschewing heavy weapons for covert action.

But the term heavy and light can apply to most weapons and roles within an army. There is heavy and light cavalry, heavy and light artillery, heavy and light tanks. In its most basic form, it is simply the same instrument which has placed a different emphasis on combat power vs. mobility.

To add onto /u/BeondTheGrave 's answer, it often also involves the role they play on the battlefield. Heavy troops typically are the slower, harder-hitting elements of your army that you deploy to break through enemy lines or otherwise to stall or repulse their advances, while light troops are faster, often used in scouting/harassing (due to their mobility) or flanking (due to their speed) roles.

If you wanted to drastically simplify and abstract it, you could say that heavy troops do a large amount of damage over a long, drawn out period of time, while light troops deal quick incisions of small amounts of damage which build up eventually.

For example, light tanks were (typically) lightly armored, fast, armed with smaller, more rapidly-firing guns, often used (for example by the Soviets) in scouting and offensive infantry support roles, while heavy tanks were larger, slower, had much heavier caliber guns, and excelled in breaching operations, pushing into urban areas, and dispatching enemy armor from long ranges.

Another example, light and heavy machine guns, typically refers to a similar situation, in which LMGs (.30 caliber was common in WW2) are smaller caliber and much easier to move around from place to place, including in the heat of combat; while HMGs (.50 caliber) are typically larger in caliber, but come with much more difficulty to transport, effectively making them static in combat due to the fact that it often times just wasn't worth it to put all the effort in to move it.

Light artillery could denote field guns or other direct-fire weaponry of smaller (under 100mm) caliber which would be attached to regiments/brigades as support weaponry for their positions, while heavy artillery typically refers to howitzers with a caliber of over 100mm which would be attached higher up the chain of command and were available to be called in/signaled for assistance from lower levels.

Light bombers are what we could think of "tactical" bombers, faster, but with less of a payload; capable of interdicting in ground engagements and using their ordnance in support of ground troops, strafing enemy formations or conducting low-level raids on enemy supply depots or administrative buildings, etc., while heavy bombers are the "strategic" bombers, the archetypal four-engined behemoths bristling with machine guns which would area-bomb from thousands and thousands of feet in the air.