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The determining a computer Architecture is complex one, starting from taking care of performance, memory requirement, computation power, availability,power consumption and the most affecting factor is now Price :-)
In Previous days computer architectures term is only referred in terms of Instruction Set design.
In reality the Architect job is not to finalize the Instruction Set for a system but also to capture all the other aspects also.
If we talk about the Instruction Set architecture(ISA), It talks about the boundary between software and hardware, there are different kind of instruction set are available now, most common are MIPS ,80x86 etc. The Instruction Set Architecture can be classified as per following points.

1. Class of ISA : The ISA can be classified on the basis of number of general purpose registers it has, like 80x86 has 16 general purpose register and 16 floating point registers. similarly MIPS has 32 General purpose and 32 floating point registers. These classes can be classified in following 2 ways. 
1. "Register Memory" ISA, it allows the access to the memory as part of instruction such as 80x86 Instruction Sets.
2. "Load and Store" It allows the access of the memory as part of load and store instructions only, such as MIPS.
2. Memory Addressing:  Most of the system will require to align the address in multiple of a number S, It is defined as "XYZ mod S" must be equal to "0" . MIPS uses the byte alignment, but for 80x86 it dose not require, but access will be faster if memory is aligned.
3. Addressing Modes:  To load , store or execute it require to fetch the data and instruction to be loaded in memory. There are different ways of accessing the memory, like registers, Immediate etc. as per MIPS design it will provide Register, Immediate and Displacement addressing modes. while in case of 80x86 it supports the following addressing modes. Absolute addressing, Indexed with displacement etc.
4. Types and Size of Operands: It plays the critical role in ISA, some of the systems supports the operands size of 8-bit (ASCII Characters), 16-its(Unicode characters), 32-bits (Word) or 64-bits(Double Word or long Integer).
5. Operations: There are different categories of operations are available like data transfer, arithmetic logical, floating point etc. The MIPS has limited set of Instructions while 80x86 has much richer instruction Set.
6. Control Flow Instructions: Most of the Instruction set architectures are support the Conditional Jump,Unconditional Jumps,Conditional branches,Procedure calls and return with value or control.
7. Encoding: There are 2 ways of Encoding fixed length and variable length. In case of fixed length It always consume the same size, while in case of variable length It consume less memory space. The MIPS instructions are 32 bit long, but the 80x86 is variable length ranging from 1 byte to 18 byte.

 If we think about the high level design of the computer architecture, it also include the computer organization  also along with ISA. The Computer organization is refer to the high level system design such as memory layout, design and there interconnection with CPU and other components. Its very much possible that Two distinct system will implement same kind of instruction set but they have different ways of caching and implementation of Instruction set pipelines like AMD Opteron 64 and Inter Pantium 4 follows the same instruction Set(80x86).