What do the letters and numbers in lens names mean across Canon, Nikon, Sigma, and other brands?

Question

Lens names often include focal length, maximum aperture, mount/format compatibility, autofocus motor type, image stabilization, and special optical/design features. The exact abbreviations vary by manufacturer—for example Nikon AF-S/DX/VR/IF-ED, Canon EF/L/USM/IS, and Sigma DG/DC/HSM/OS/APO/Macro.

How can I interpret these codes in a practical way, and which parts are universal versus brand-specific?

user124 · Accepted Answer

^{Note: Because of the community-wiki nature of this question, this accepted answer became really long, and difficult to edit and keep current as lens designations evolved. The historical answer has been broken up into individual answers per lens brand, with links to each of the lens brand answers below.}

Descriptions by Lens Make

Brand Lenses

All major camera manufacturers offer their own line of lenses. Such lenses tend to follow the most stringent quality guidelines, and often come with a price premium.

Off-Brand Lenses

Most off-brand lens manufacturers make lenses that fit many types of bodies, including Canon, Nikon, etc.

Samyang (also branded as: Pro-Optic; Rokinon; Bower)
Sigma
Tamron
Tokina

Descriptions by Lens Features

If you know what feature you're looking for in a lens (cropped-frame designation ultrasonic motor, low-dispersion elements, image stabilization, etc.), and want to know what each of the brands call that feature, the following answers are organized by lens feature.

Descriptions of Lens Features