BQN's three operation types are the function, 1-modifier, and 2-modifier.
In general, an operation is called by passing in inputs, and returns a result value. The inputs and result can have any type. Since BQN isn't a pure functional language, the operation might also have side effects: it can modify the values of variables, perform program input or output, or call other operations with their own side effects.
This page deals with types, not syntax. Expressions with a function or modifier role don't have to yield a value of that type when run. However, primitives and blocks do have roles that match their types.
A function has one or two inputs called arguments. The general layout is 𝕨 Fn 𝕩, with an optional left argument 𝕨 and a non-optional right argument 𝕩. The number of arguments is called its valence, and functions are naturally ambivalent, allowing one or two arguments. When called with one argument—𝕩 only—it's monadic, and when called with two it's dyadic. More arguments, or a variable number, should be handled by using a list argument; destructuring headers can be useful in this case.
Functions can be primitives or blocks (or system functions), but there are also two kinds of compound functions: derived functions that consist of a modifier and its operands, and trains. Tacit programming refers to code written without blocks, so of course it uses compound functions heavily.
↗️3 + 4 # Primitive function 7 {𝕩+𝕩} 4 # Block function 8 +˜ 4 # Derived function 8 (⊢+÷) 4 # Train 4.25
Compound functions have some differences with blocks, most importantly that blocks can express mutation while a compound function can't have side effects unless one of its constituent functions or modifiers does. More subtly, compound functions match when they have the same composition (much like lists) while blocks must be the same instance to match. A function's composition can also be inspected directly with •Decompose (spec).
While normally functions are just called, some primitives might try to infer properties of functions, which necessarily involves inspecting their definitions. The identity value used by reductions and the results of Undo and Under rely on inference.
There are two modifier types, separated for syntax reasons: 1-modifiers follow the layout 𝔽 _mod and 2-modifiers follow 𝔽 _mod_ 𝔾. The values 𝔽 and 𝔾 are called operands. There aren't any compound modifiers, so modifiers are always primitives or system-provided, or blocks. A primitive is a 1-modifier when it's written as a superscript like ˘ or ˝, and a 2-modifier when it has an unbroken circle like ∘ or ⍟ (not ⌽ or ⍉).
+⎉3 # Primitive 2-modifier (deferred) +⎉3 2{-𝕗} # Block 1-modifier ¯2
In general, a modifier call works just like a function: inputs out, result in. Syntactically, a modifier call expression has a function role, but that doesn't affect execution. However, one kind of modifier is more strict: a deferred modifier doesn't evaluate anything when called, but returns a derived function. When the derived function is finally called, the modifier's definition determines what happens. Primitive modifiers are always deferred, and a block modifier is deferred if it includes arguments, either in the header or with 𝕨, 𝕩, or 𝕤 in the body.