Read at your own risk!
Aspects are the "flavors" of magic, the forms of vis that can be absorbed into wands, captured in jars, and used in various pursuits. There are 6 Primal aspects, which combine in various ways to produce 44 Combined Aspects, for a total of 50.
Here is a spoileriffic list of the aspects and their compositions. Spoilers!
Uses
The Primal aspects of vis can be drawn into wands and stored. Once in the wand, these can be used in various magical craftings. With warded jars, essentia of combined aspects can also be stored for use. Neither vis nor essentia can be combined aspect-wise (for example, you cannot combine Terra and Aqua vis to make Victus), but with appropriate devices (centrifuge or even crucible), essentia of a compound aspect can be broken up into its components. Vis can be drawn from nodes and various devices, but small amounts are also scattered every time a mob is killed -- if a nearby player has a wand in their hotbar, the wand will try to absorb these, subject to its capacity. Essentia can only be produced by breaking down material objects.
In addition to vis, the Aspects also affect research -- research points are gained in particular aspects (mostly from the Thaumometer), and used to research things relevant to their nature. In contrast to the magical uses, research points for compound aspects cannot be separated, but points can be combined aspect-wise.
Here is a spoileriffic routine for discovering all the aspects, and an extensive guide to research. Spoilers!
Aspects and Items.
Each item (also mobs, blocks, and perhaps entities) also has one or more aspects. These can be converted to essentia by a Cauldron or Alchemical Furnace, then used for other processes.
Below is a list of the best items to get aspects wholesale. Some entries also mention a type of essentia they can be centrifuged from, which is itself easy to get. Two items are particularly important for advanced players, as between them, they renewably provide the essentia aspects that are most needed in large quantities: Charcoal provides Ignis for the Infernal Furnace, and Potentia for the Thaumostatic Harness. Netherwart provides Praecantio for making thaumium and many other enchantments, and Herba for the Lamp of Growth. (This leaves the Arcane Bore, whose thirst for Perditio can be met with cobblestone or cactus.)
A tip for tools, boats and other things that don't stack: The arcane workbench holds items in place even when the GUI is not open. Arrange enough source materials for the desired quantity, take as many as your inventory will hold, load them into a hopper connected to the arcane furnace, and keep collecting from the workbench as you free up inventory space.
All types of essentia can also be obtained from mana beans and, to a lesser extent, ethereal essence.