How does a processor 'know' what the different commands mean?
I'm thinking of assembly level commands like MOV, PUSH, CALL, etc...
When a computer interprets assembly level instructions, these instructions are turned into their binary equivalents for the CPU to read. When the CPU executes the instructions, it interprets the opcode part of the instruction into individual "microprograms", containing their microcode equivalents. Just so you know, a full assembly instruction consists of an opcode and any applicable data that goes with it, if required (e.g. register names, memory addresses).
Microcode instructions are extremely low-level (more so then assembly), and control the actual digital signals which control the flow of logic in the microprocessor. For example, one microcode instruction could update a condition code register flag with a new value, or connect a CPU register with one of the ALU units. More complex tasks are possible, but this shows you the general idea of what microcode is used for.
The general flow from compilation to execution is as follows. The assembly instructions are assembled (turned into their binary equivalent 0s and 1s, or from now on, logic signals). These logic signals are in-turn interpreted by the CPU, and turned into more low-level logic signals which direct the flow of the CPU to execute the particular instruction. This can take one or more clock cycles, depending on the processor's architecture and design (most processor reference manuals tell you how many clock cycles it takes to execute a particular instruction, like this one for example).
All of this is done with hard-programmed microcode (physically embedded within the processor in some kind of ROM, set during manufacturing), which directs the flow through actual low-level logic gates. This provides an interface between the abstract assembly instructions and the physical electrical logic in the processor.
So, in summary, processor instructions are assembled and loaded by the processor. The processor will then use these instructions to look up the microprogram (in the form of microcode) corresponding to that particular instruction, which is what "actually" executes the instruction. Once the microcodes for the particular instruction have been executed (which can take one or more clock cycles), the processor executes the microcode to fetch the next instruction, and the cycle repeats.
The processor doesn't really 'know' what the commands are. The commands are just binary patterns that cause the processor to do what we interpret the commands to mean.
For example, an ADD-R1-into-R2 operation will cause registers 1 and 2's values to reach the ALU (arithmetic and logic unit), cause the ALU to use the output of the adder instead of the various other stuff, and cause the output of the ALU to replace the value in register 2. There are simple logic circuits to achieve all of these things (multiplexer, adder, counter, ...), although real processors use very complicated optimizations.
It's sortof like you're asking how a car knows to slow down when you press the brakes. The car doesn't know, the brake pedal just happens to indirectly control how hard pads are pressed against the wheels.
Take, for example, the instruction that tells an x86/IA-32 processor to move an immediate 8-bit value into a register. The binary code for this instruction is 10110 followed by a 3-bit identifier for which register to use. The identifier for the AL register is 000, so the following machine code loads the AL register with the data 01100001.
This binary computer code can be made more human-readable by expressing it in hexadecimal as follows
Here, B0 means 'Move a copy of the following value into AL', and 61 is a hexadecimal representation of the value 01100001, which is 97 in decimal. Intel assembly language provides the mnemonic MOV (an abbreviation of move) for instructions such as this, so the machine code above can be written as follows in assembly language, complete with an explanatory comment if required, after the semicolon. This is much easier to read and to remember.
In other words, when you 'assemble' your assembly program, your instructions such as
MOV AL, 61h
are converted to numbers, which the CPU associates a special meaning and then acts accordingly.
Also check out the course notes from CS152: Computer Architecture and Engineering at UC Berkeley, a course in which students implement a CPU.
If you google for "home-built cpu" you will find many goodies.
At the extreme most lowest level, all the CPU can do is add. From addition, it can subtract, multiply and divide (seeing as these are just addition in a different manner). The CPU uses this to move data around in memory by applying the additions to memory addresses.
Keep in mind though that this is at the lowest level possible. The CPU does in fact "understand" certain commands, in the form of microcode. See Breakthrough's answer, it's very well written.
I've given a related answer at programmers.stackexchange.com, see How do computers work? where I walked briefly over everything from the ground up about how computers goes about interpreting instructions to moving electrons.