My system, which my small community of friends uses, is potentially straightforward. (Maybe.)
Regular blocks are simply an X by X.
Plates are an X by X plate.
Studless plates are an X by X flat plate.
Bionicle pieces are Bionicle (set name) (piece use)
Minifigure pieces are (figure) head/torso/leg/waist/arm/hand/hair/hat
Gears are (tooth count) (special features)
1/2 gears are flat (tooth count)
Technic pieces are called by function, e.g. Worm, or by set, e.g Power Miners Claw
Technic axles are exactly as printed in the set instructions
Technic bars are X hole bars
Curved bars include the angle as well, e.g. cross, 6 holes, 45, 45, 2 holes, cross
Accesories for minis are called by their apparent look
Rubber pieces describe their holes, e.g. two crosses
Studs are called studs
Sloped pieces are X by X by X (slope/multislope) (with/without studs)
Circular plates are X by X circular (flat?) plates
Printed pieces are called by their obvious type, e.g. Milk carton, Dragon-printed door plate
NXT set pieces are called by the names given in the set
Racheting pieces are X by X with X ratchets
Claw-and-bar pieces are X by X claw/bar (plate?) s
Wheels are cross-axle/tpin/pin wheels, radius X in studs
Turntables are X by X turntables
Hinges are (plate?) hinges, multiple types inapplicable
Curved pieces are (studs) (plate?) curves
Sloped plates are (studs) angle plates
Motors are (windup?) (spec) motors
Remotes are called by their set specs
Tyres are described by what they fit, or as treads
Technic pins (called tpins for wheel purposes,) are (friction) X-length pins (with cross adapter?)
Technic bars with studs are X-stud bars
Strange miscellaneous pieces are described as thoroughly as possible, including source set
Launchers describe what they shoot, e.g. rubber-head missile launcher
Chains are described by length in studs
Vertical rotators are called that, X by X top
Animals and microfigures are called by their obvious identity
Capes are short, regular, or mini-specific, e.g Ringwraith cape
Feel free to add extra thoughts in the comments, or reference people here from non-meta