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MHz (Megahertz) is the number of clock cycles every second. This is usually determined by a crystal oscillator on the board somewhere. For instance, if your processor or memory runs at 20MHz, it gets 20 million clock cycles every second. What it does with this depends entirely on the chip.
Some processors will perform one instruction for every clock pulse. Some will take several clock pulses for some (or all) instructions. For instance, the simple x51-type processors perform one simple instruction in 12 clock cycles. They call these 12 clock cycles one machine cycle. More recent varients may only take 4 clock cycles for each machine cycle, thus running 3 times faster (doing 3 times more instructions) on the same clock frequency.
This is where MIPS comes in. MIPS is Millions of Instructions Per Second. That is, how many actual instructions are performed - it may be different from the clock frequency as described above.
For instance, if an old x51 cpu has a clock of 24MHz, and takes 12 clock cycles per machine cycle, it can do 2 MIPS (24MHz/12 cycles). Even then, some instructions may take more than one machine cycle to perform so an actual program may run at less than this. If a more modern version of the core is used, that uses 4 clock cycles per machine cycle, it will process instructions at a rate of 6 MIPS (24/4) - 3 times faster for the same clock.
If comparing similar types of cpu, MIPS gives a more meaningful comparison of speed than MHz.
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