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Here is a tested and working example of a state machine. Say you are on a serial stream (serial port, tcp/ip data, or file are typical examples). In this case I am looking for a specific packet structure that can be broken into three parts, sync, length, and payload. I have three states, one is idle, waiting for the sync, the second is we have a good sync the next byte should be length, and the third state is accumulate the payload.
The example is purely serial with only one buffer, as written here it will recover from a bad byte or packet, possibly discarding a packet but eventually recovering, you can do other things like a sliding window to allow for immediate recovery. This would be where you have say a partial packet that is cut short then a new complete packet starts, the code below wont detect this and will throw away the partial as well as the whole packet and recover on the next. A sliding window would save you there if you really needed to process all the whole packets.
I use this kind of a state machine all the time be it serial data streams, tcp/ip, file i/o. Or perhaps tcp/ip protocols themselves, say you want to send an email, open the port, wait for the server to send a response, send HELO, wait for the server to send a packet, send a packet, wait for the reply, etc. Essentially in that case as well as in the case below you may be idling waiting for that next byte/packet to come in. To remember what you were waiting for, also to re-use the code that waits for something you can use state variables. The same way that state machines are used in logic (waiting for the next clock, what was I waiting for).
Just like in logic, you may want to do something different for each state, in this case if I have a good sync pattern I reset the offset into my storage as well as reset the checksum accumulator. The packet length state demonstrates a case where you may want to abort out of the normal control path. Not all, in fac