With buildings you produce units. These units build up your army and perform essential roles such as attack and base building. Units generally take supply which constrains the number of them that can be used at a time.
Several units have additional abilities.
Abilities are attributes of a unit that differentiate their potential uses from those of other units. These include Flying and Detector and other similar traits. These include manually activated, conditionally activated, auto-cast, passive, or even permanently active effects. Some abilities can be switched to autocast, so they will be activated automatically when certain conditions are met.
Spells are abilities that can be activated manually. Most spellcaster types use up a certain amount of stored Energy, which starts out at 50 (and can be upgraded to 75 in the case of certain units) and slowly builds to 200 at a rate of about 1 per game second. Generally speaking, spells in StarCraft II radically change conditions of units or the map or fulfill other essential functions and have no cooldown time, although they do have a cast time before they are activated, sometimes during which the caster is helpless.
Peon is a general term for Worker. They are produced from the main base (or Townhall) for each race and cannot be made from any other buildings. They are the only units that harvest resources and are also essential for constructing buildings. They possess a weak melee attack that can not be upgraded but can help in critical early-game situations (or all-in rushes).
Upgrades improve the functionality of units by improving their basic statistics or granting them additional abilities. Upgrades in StarCraft apply to all units under their effect, even those that already have been made before upgrades have been finished.
All upgrades take a specific amount of resources and time to complete before they are researched and are researched at associated technology buildings. They usually require Vespene Gas which often factors in significantly how fast they may be available. There are sets of upgrades that in general apply to all races.
Weapon / Armor Upgrades increase units' attack or armor value by differing amounts and may apply to one unit or a group of units. Basic offensive units deal their damage in two parts and so are upgraded twice by one damage. Units with a high basic damage gain more than one point of damage per upgrade. Armor upgrades always have the value one. The upgrade factor is given in parentheses after the basic damage, e.g. 15 (+3). Typically, specific two racial buildings are used for upgrades to most units. They are the Engineering Bay and Armory, the Evolution Chamber and Spire and the Forge and Cybernetics Core.
Most spellcasters can be upgraded to spawn with 75 Energy instead of 50.
Transformations are "upgrades" applied to singular targeted buildings or units that change their functionality (referred to in-game as mutating, upgrading, or building buildings or units). They may require the expending of resources. Examples include the Command Center to Orbital Command upgrade,
Corruptor to Broodlord upgrade, and the Gateway to Warpgate ability.