Alchemy is the study of materials and how to transform them with magic. Chemistry also transforms materials, but without the aid of magic. They are considered a single specialty in the Nolaria Campaign. Alchemy enables the creation of potions, powders, compounds, and various magical materials. The following descriptions use potions as an example, but they apply to the creation of any alchemetic item.
Formulas
For most magic potions there is generally one known way to make it, though subtle variations of potions may have different formulas. The formula is a set of instructions that tell the alchemist what materials, tools, and processes he needs to make the potion. The formula is very exact and demanding of precision. Deviations will result in potions that don’t work -- at best! These formulas tend to be very rare and jealously guarded. As such, they are quite valuable and can cost a lot to obtain.
Materials
A formula may call for almost anything in the construction of a magic item, but some materials are more common than others. Sometimes materials must be treated, worked, or processed before it can be used in a potion. Although it can take many years of studying the properties of materials, an alchemist can make quite a bit of money by supplying materials to others who need it, such as artificers. Because they never quite know what may be needed, alchemists tend to acquire and collect various materials, especially rare, unique, or magical ones. An essential part of any well equipped alchemy lab is a good stock room of materials.
Tools
Many tools can be used in the creation of potions and alchemetical substances. Some of these are common tools: caldrons, glassware, mortar and pestle, etc. Others are made for a specific purpose or to handle some special task.
More rarely, tools are magic items themselves and must be obtained or made before a potion calling for them can be created. In some cases, magic alchemy tools or devices just make production or processes simpler (or safer) and are not absolute requirements for a formula. Making gem dust is a good example. It's easy to shatter a gem using a spell of the same name, but making sure the pieces don't go all over the place and the end result is a uniform dust is what alchemy does.
Processes
In the course of creating alchemetical materials or potions, a specific process is sometimes called for. This may be a means of treating some piece of material, a means of combining ingredients, or instructions for something to be made. Some of these processes can be learned separately and are commonly used. Common processes include grinding, filtration, fermenting, heating, cooling, and many others.
Alchemy in 5e
Alchemy in the 5e rules system is associated with alchemy tools and certain types of backgrounds (Artisan, Folk Hero, Sage), classes (Monk). This implies that any character class would work alchemy and to some degree this is reasonable. However, there are some processes (etc) that require spell casting abilities. For this and other reasons, alchemy was strongly associated with Mages in AD&D2. Non-spellcasters may run into limits that are difficult to overcome.
The PHB says, "Any character proficient with alchemy tools knows how to use all of its component parts." While this vastly simplifies the play of alchemists in 5e, much of the depth is lost. While some alchemy may be practices by just buy a set of tools (Portable Alchemy Lab), Nolaria defines Alchemy as a specialty.
Proficiency with Alchemy reveals more information about materials using an Arcana check. On investigation checks, it will give clues chemicals (etc) used in the area.
Becoming an Alchemist
There is a bit of a conflict between the way 5e handles alchemy and the Alchemy Specialization. In 5e, characters can attempt simple alchemy from first level. To some degree, this represents the difference between on-the-go alchemy versus work in a lab. Alchemy can be incompatible with a wandering lifestyle. Still, there are some things that can be done in the field. The Nolaria Campaign defines Alchemy as a skill that improves with practices and tools.
From the PHB, a Monk, Artisan, Folk Hero, and Sage may have an artisan tool, which includes alchemy tools. This conveys a set of portable alchemy tools and advantage on DC checks.
Background details
It is possible to learn alchemy with out the background describe above. Provided they can be found, anyone may purchase alchemist tools. With the kit comes recipes for starting a fire, extracting oil (processing plant materials), making soap (mixing lye and fats). Some other recipes are common, others more difficult.
Advancing
Advancing in a specialty is a bit different than collecting experience points to level up. Rather, rough levels of knowledge/skills are defined and the DM decides when the character reaches a new level. Advancement in alchemy is an combination of skills, knowledge, experience, and insight.
When evaluating a character for advancement in alchemy, the DM will look for these things:
Number of books in library or access.
Sophistication of lab and equipment.
Knowledge of materials.
Knowledge of processes.
Collection of archetypes.
Success in creating alchemetic materials.