Well, each country did have a different structure but remained similar for most of the eighteenth century. Let's start with the British.

British Army

The lowest rank is of course the private. They made up the majority of the regiments and army. 38 privates made up a company usually, at least by the American War of Independence. They wore the cheaper madder redcoats. Privates had no command whatsoever. Any that proved themselves worth would be promoted to corporal with only 3 a company. A rank above private and only had the privilege to drill and train new recruits. They wore the same coat but with either a white cord or silk epaulette - depending on the year of the war. They might have control of a squad every now and then, but rarely. The next rank is the sergeant or the non commissioned officers.

Sergeants did have command in various degrees in the army. Often of a platoon and again in drilling. They also made sure the lines remained dressed in parade and in battle as the rank leaders. They dressed a bit different too. Sergeants had a mock scarlet coat with a sash. The sash was red with the facing color in the middle. For most regiments they wore the sash around their waist, Highland regiments would have theirs slung from the shoulder to hip. A sergeant-major did exist in the eighteenth century. Only one per regiment and they had the same privileges of command as a sergeant but also dealt with command staff a lot more since they were often the most senior of sergeants. In earlier years, sergeants carried a halberd but by the AWI, they carried a sergeants fusil instead. Sergeants would be one of the ways to relay commands from officers/drummers on the battlefield as well. Each company had two sergeants which divided up the company into two platoons.

Officers ranged from the lowly ensign to the colonel. This doesn't include the staff officers that would usually not take field command of regiments directly. Ensign comes from the term of flag-bearer in a way. A lot of the times in the eighteenth century, young men made up the numbers of ensigns. Sometimes as young as 14 years old. It was a decent way for a family of wealth to get a son in the army as an officer without too much expense. Commissioned officers differed from those of rank and file, and the non-commissioned officers as the man had to actually buy his commission. The lowest at ensign and highest at lieutenant colonel, each getting most expensive as the rank climbs. The royal artillery called them second lieutenants, likewise with fusilier regiments. Ensign barely had command in the company. The other two officers of the company, lieutenant and captain, would have more control. Officers had the nicer fine cloth of a scarlet coat. The epaulettes on one or both shoulder depended on rank and the metal depended on the regiment. Some had silver, some had gold.

The flank companies were kind of outliers with the offices. Battalion companies had the ensign, lieutenant, and captain officers (exceptions for Royal artillery and fusiliers) while flank companies often had two lieutenants and a captain. Flank companies were either the grenadier or light company of the battalion. The army wanted experienced officers for flank companies, which ensigns did not qualify for. The odd rank of captain-lieutenant also existed. It was usually reserved for the 1st company of the battalion which the colonel "commanded" on paper. As expected, the colonel cannot always command the company or regiment directly on the field so the privilege will go to the captain-lieutenant. What was unfortunate for the captain-lieutenant was that he is paid the same amount as a lieutenant but has the responsibilities of a captain. You wanted experienced officers in the flank companies since they will be be doing some hardcore fighting when the battle begins. The types of fighting needs competent officers at the helm, not something for the ensign to attempt to tackle by himself.

Staff officers began at the rank of major. Major's did hold command of a company but may sometimes lend it to a lieutenant or captain-lieutenant in his stead. Staff officers had epaulettes on both shoulders and scarlet coats. Majors usually had positions in the brigade as a commandant or aide de camp to a a general. Some majors did take the position of adjutant but a lieutenant would be more common in position. Generals commanded the army but also had divisions with the ranks. The brigadier general rank would only last the duration of the war and led the commanding of the brigades. Two to four regiments would make up a brigade. Colonels of regiments could also be brigadier generals at the same time. Allan MacLean of Torloisk commanded the 84th regiment of foot while also holding the rank of brigadier general in Canada during the AWI.

Speaking of holding ranks. Other officers could also hold two ranks. One in the army and one within the regiment. So one could be a major in the army but have a rank of captain within a regiment. It depended on the vacancies of the regiment. A battalion can only have one major as a company commander. Thus, an incoming major would have to take command as a captain, but still held the rank of major in the army. A bit confusing isn't it?

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Sources:

J. A. Houlding, Fit for Service: The Training of the British Army 1715-1795

Steven M. Baule and Stephen Gilbert, British Army Officers Who Served in the American Revolution