Their specification and format are laid out in the instruction set architecture (ISA) of the processor in question (which may be a general CPU or a more specialized processing unit). Apart from the opcode itself, an instruction normally also has one or more specifiers for operands (i.e. data) on which the operation should act, although some operations may have implicit operands, or none at all. There are instruction sets with nearly uniform fields for opcode and operand specifiers, as well as others (the x86 architecture for instance) with a more complicated, varied length structure. 
Depending on architecture, the operands may be register values, values in the stack, other memory values, I/O ports, etc., specified and accessed using more or less complex addressing modes. The types of operations include arithmetics, data copying, logical operations, and program control, as well as special instructions (such as CPUID and others).
Assembly language, or just assembly, is a low-level programming language, which uses mnemonics, instructions and operands to represent machine code. This enhances the readability while still giving precise control over the machine instructions. Most programming is currently done using high-level programming languages, which are typically easier to read and write. These languages need to be compiled (translated into assembly language), or run through other compiled programs.
Software instruction sets
Opcodes can also be found in so called byte codes and other representations intended for a software interpreter rather than a hardware device. These software based instruction sets often employ slightly higher-level data types and operations than most hardware counterparts, but are nevertheless constructed along similar lines. Examples include the byte code found in Java class files which are then interpreted by the Java Virtual Machine (JVM), the byte code used in GNU Emacs for compiled LISP code, .NET Common Intermediate Language (CIL), and many others.